It is difficult to overturn an American election result, as President Trump was reminded on Friday, when his campaign lost in courts in Michigan and Pennsylvania and dropped a challenge in Arizona.

A losing candidate who is within striking distance — say a few hundred votes — might get lucky in a recount. Beyond that, he or she would need to show systemic fraud on a large scale.

Mr. Trump has shown nothing like systemic fraud in any of the lawsuits, 16 and counting, that his campaign and allies have filed since Election Day as they seek to block certification of Joseph R. Biden Jr.’s victory in the Electoral College. The allegations of fraud are a smattering of unverified accusations about the voting or counting process, usually directly affecting too few ballots to change a state’s results.

Other lawsuits in the batch depend on the theory that differences among counties in verifying or including absentee ballots amount to a violation of a voter’s constitutional right to equal protection. Given how decentralized the voting system is in the United States, that argument has yet to get legal traction after this election.

Here is a rundown of the cases and their statuses.

Michigan

Costantino v. Detroit

Filed: Nov. 9

Claim: The plaintiffs were two poll challengers — representatives of the Republican Party allowed to monitor the vote counting — who alleged fraud and misconduct during the vote count last week at the TCF Center in Detroit. The lawsuit asked the court to block certification of the election results in Wayne County (which includes Detroit), where more than 867,000 people voted.

The plaintiffs claimed that election workers were instructed “to not verify signatures on absentee ballots, to backdate absentee ballots, and to process such ballots regardless of their validity.” They presented five affidavits from other poll challengers who said, among other things, that they saw election workers enter Jan. 1, 1900, as the birth date of many absentee voters. The plaintiffs offered no corroborating evidence for the affidavits. They also falsely stated that Republican poll challengers could not observe the count.

Context: The plaintiffs misunderstood the process for verifying and tabulating absentee votes, according to Chris Thomas, the state director of elections in Michigan for 36 years before his retirement in 2017, who submitted an affidavit for the defendants, the City of Detroit and the Detroit Election Commission. He explained that election workers entered Jan. 1, 1900, as a placeholder when the computer system required them to type in a birth date but they could not access that information in the voter file.

Mr. Thomas also said that ballots received on Nov. 3 were entered on Nov. 4 after election workers discovered that they had not previously been fully processed. Finally, he explained that Detroit allows 134 challengers each from the Republican and Democratic Parties, and once those limits were reached, access for a new challenger was allowed when someone left the hall.

Status: On Friday, Chief Judge Timothy M. Kenny of Wayne County Circuit Court denied the petition from the pro-Trump side. “It would be an unprecedented exercise of judicial activism for this court to stop the certification process of the Wayne County Board of Canvassers,” Judge Kenny wrote. “Plaintiffs’ interpretation of events is incorrect and not credible.”

Donald J. Trump for President Inc. v. Benson

Filed: Nov. 11

Claim: This is a federal suit that repeats the pending state-court claims of fraud and misconduct in Costantino v. Detroit. The Trump campaign is similarly seeking to block Michigan’s certification of the vote.

The decision to file a second case suggests that the Trump campaign is “trying to try the same case in multiple courtrooms hoping that somebody will provide an endorsement of their baseless conspiracy theories,” said David Fink, an outside lawyer for the city of Detroit.

Context: The plaintiffs submitted more than 230 pages of affidavits from Republican poll challengers. But they described isolated grievances and perceived irregularities, not systemic fraud. “I felt intimidated by union people who were staring at me,” one poll challenger complained. Another said the loud volume of the P.A. system was distracting. A third said a Democratic poll worker told her to “go back to the suburbs, Karen.” A fourth found it suspicious that most of the few dozen ballots he saw counted that were cast by members of the military were votes for Mr. Biden.

Status: Pending.

Donald J. Trump for President Inc. v. Benson

Filed: Nov. 4

Claim: During the ballot counting in Michigan, the Trump campaign sued to stop it. In an affidavit, one poll watcher said another, whom she did not name, told her that she was told by still other poll workers to change the date on which a ballot was received.

Context: This was one of a number of cases the Trump campaign filed in key states to try to stop the counting as the results swung in favor of Mr. Biden.

Status: At a hearing, Judge Cynthia Stephens of the Michigan Court of Claims characterized the affidavit as, “‘I heard someone else say something,’” and said, “Tell me how that is not hearsay. Come on now!” Judge Stephens dismissed the suit on Nov. 5. The Trump campaign’s appeal of her ruling was denied because it lacked the required attachments.

Stoddard, et.al. v City Election Commission of the City of Detroit

Filed: Nov. 4

Claim: A Republican poll challenger and the Election Integrity Fund, a nonprofit organization, sued to stop election workers in Detroit from hand duplicating ballots that could not be read by a machine. The suit also sought to delay certification of the results.

Context: This was an early effort to suggest that mail-in ballots were being handled to benefit Mr. Biden.

Status: Judge Kenny dismissed the suit on Nov. 6. “Plaintiffs fail to identify the occurrence and scope of any alleged violation,” he wrote.

Pennsylvania

In re: Canvass of Absentee and Mail-In Ballots of Nov. 3, 2020, General Election

Filed: Nov. 10

Claim: On behalf of a voter, the Trump campaign appealed the decision by the Philadelphia County Board of Elections to count five categories of mail-in ballots. Pennsylvania law provides that voters must sign and “fill out” an outer envelope (as well as include an inner security envelope when they return their ballots). This suit challenges a total of 8,349 ballots with outer envelopes that were signed but lacked other information, like the date or the voter’s printed name or street address.

Context: The number of ballots challenged in this suit for minor errors would not be enough to change the election result.

Status: On Friday, Judge James Crumlish of the Court of Common Pleas denied all of the challenges to all five categories of ballots.

Pirkle v. Wolf

Filed: Nov. 10

Claim: Four voters sought to block all votes from Philadelphia, Montgomery, Delaware and Allegheny Counties from being included in the state total, claiming that the state violated the right to equal protection by allowing differing absentee balloting practices among counties.

Before the election, Pennsylvania’s secretary of state, Kathy Boockvar, advised county election officials that “curing” absentee ballots (the term for fixing mistakes like a missing signature) was permitted but not required. As a result, local practices differed to a degree. This suit claims that it was unconstitutional to allow some but not all Philadelphia voters to cure their ballots. The suit also cites Delaware County for allegedly giving in-person ballots to voters who were recorded as having received mail-in ballots without requiring them to sign the registration book at the polls. In Allegheny County, according to the suit, voters were required to vote provisionally when records showed they had requested an absentee ballot but not cast it.

Context: The election system in the United States is highly decentralized and gives considerable authority to state and county officials over how to run elections. As a result, it is common for states to allow some variation in local election practices and to allow voters to cast provisional ballots in a variety of circumstances.

Status: On Thursday night, the law firm representing the Trump campaign, Porter Wright, asked to withdraw. Previously, the plaintiffs asked Judge Matthew Brann of the United States District Court for the Middle District of Pennsylvania to consolidate this case with the next one below.

Donald J. Trump for President v. Boockvar

Filed: Nov. 9

Claim: The Trump campaign is seeking to block the certification of the Pennsylvania election, alleging fraud in mail-in balloting, insufficient access for poll observers and varying procedures for curing ballots among different counties.

The plaintiffs said “Democratic heavy” counties allowed voters to cure their absentee ballots while “Republican heavy” counties did not. They also said ballots were processed in Allegheny County and Philadelphia while poll observers were too far away to see what was happening.

Context: On Election Day, Republican lawyers acknowledged in court that the party’s observers were present at the Pennsylvania Convention Center in Philadelphia when votes were being counted. “I’m sorry, then what’s your problem?” Judge Paul S. Diamond of the United States District Court for the Eastern District of Pennsylvania asked. He granted a modest accommodation, ordering the city election commission to allow poll observers to move closer to the counting.

Status: Judge Brann scheduled oral argument for Nov. 17.

Donald J. Trump for President Inc. v. Bucks County Board of Elections

Transition: Latest Updates

Filed: Nov. 9

Claim: The Trump campaign and others appealed the decision of the Bucks County Board of Elections to count 2,175 ballots that lacked the voter’s printed name or street address, or the date of signing. The suit also challenged the inclusion of 76 ballots that arrived in unsealed inner envelopes or with markings on them.

Context: Like a number of other suits, this one involves a relatively small number of votes that would not change the outcome.

Status: Judge Robert O. Baldi of the Court of Common Pleas scheduled a hearing for Nov. 17.

Donald J. Trump for President Inc. v. Montgomery County Board of Elections

Filed: Nov. 5

Claim: The Trump campaign and others sued the Montgomery County Board of Elections for notifying voters before Election Day to allow them to fill in missing information on ballot envelopes.

Context: About 600 ballots are at issue. At a hearing, Judge Richard P. Haaz of the Court of Common Pleas asked if the suit included any accusations of fraud. When pressed, the lawyer for the plaintiffs said no.

Status: On Friday, Judge Haaz denied the petition to disqualify the ballots. “Voters should not be disenfranchised by reasonably relying upon voting instructions provided by election officials which are consistent with the Election Code,” he wrote.

Donald J. Trump for President, Inc. et al v. Boockvar et al

Filed: Nov. 4

Claim: The Trump campaign and others challenged guidance that Ms. Boockvar provided to counties before the election, allowing them to tell absentee voters they had until Nov. 12 to provide missing proof of identification.

Context: Pennsylvania law normally sets a deadline for six days after the election, Nov. 9 this year, for providing missing voter ID. Ms. Boockvar, the secretary of state, extended the deadline to Nov. 12 in light of a decision by the Pennsylvania Supreme Court to allow absentee ballots to be counted if they were received by Nov. 6. The extension Ms. Boockvar allowed applied to only a small number of ballots with missing ID that arrived after Nov. 3.

Status: On Nov. 12, Judge Mary Hannah Leavitt of the Commonwealth Court ordered this narrow category of ballots to be excluded from the state count. None of these ballots had yet been included. This is the Trump campaign’s only win so far in postelection litigation.

Republican Party of Pennsylvania v. Boockvar

Filed: Nov. 6

Claim: The Pennsylvania Republican Party asked the Supreme Court to block the counting of ballots received between the end of Election Day and Nov. 6.

Context: In September, the Pennsylvania Supreme Court ruled that ballots could be counted if they were postmarked by Nov. 3 and received by Nov. 6. Republicans then asked the U.S. Supreme Court to block the Pennsylvania court’s ruling. In October, in a 4-to-4 split, the justices let the state court ruling stand. Pennsylvania’s attorney general, however, ordered the late-arriving ballots segregated from the rest of the count.

On Nov. 6, Justice Samuel A. Alito Jr. also ordered the late-arriving ballots segregated. In the end, there were fewer than 10,000 such votes. They have not been included in the reported state total. By themselves, those ballots would not affect the outcome in Pennsylvania.

Status: On Friday, the U.S. Court of Appeals for the Third Circuit ruled that the plaintiffs did not have standing to stop the counting of the late-arriving ballots. The Supreme Court has yet to say whether it will hear the Republicans’ appeal.

Arizona

Donald J. Trump for President Inc. v. Hobbs

Filed: Nov. 9

Claims: The Trump campaign claimed that some ballots in Maricopa County were filled out with Sharpie pens. A handful of voters and a poll observer said they saw bleeding ink on these ballots, causing an “overvote” on the other side, and that poll workers were instructed to press a green button on the scanner to accept the votes despite the error. The suit seeks to prevent the supposed “overvotes” from being included in the tally until they are reviewed.

Context: The suit involves only 191 votes for president. At a hearing on Thursday, when one witness was asked if she had a basis to believe her vote was not counted, she answered, “Uh, I’m not sure.”

Status: On Friday, the Trump campaign dropped the suit.

Nevada

Stokke v. Cegavske

Filed: Nov. 5

Claims: Two voters, a poll observer and elected officials, backed by the Trump campaign, claimed there were problems with an automated signature verification machine in Clark County. One of them, Jill Stokke, said at a Trump campaign news conference, “I went to vote and was told I already voted.”

Context: The Clark County registrar of voters, Joe Gloria, investigated Ms. Stokke’s complaint and said that the signature on her absentee ballot appeared to match the one in the voter file, but gave her the option of challenging that vote and casting a provisional ballot. Ms. Stokke refused.

Status: Judge Andrew Gordon of the United States District Court for the District of Nevada denied the plaintiff’s request on Nov. 6 for lack of evidence.

Georgia

Brooks v. Mahoney

Filed: Nov. 11

Claim: Four Republican voters sued over the process of absentee vote counting in several Georgia counties, seeking to exclude all of the votes cast in them from the state total.

The plaintiffs cited isolated instances of fraud. One voter, for example, said his absentee ballot was recorded even though he voted in person. The plaintiffs also claimed that several counties appeared to have more registered voters than their interpretation of census data would suggest was plausible. They cited a study that estimated from past survey data, but without proof, that thousands of noncitizens voted for Mr. Biden.

Context: Georgia is in the middle of a recount.

Status: The suit is pending in the United States District Court for the Southern District of Georgia, Savannah Division.

In re: enforcement of election laws

Filed: Nov. 4

Claim: The Trump campaign sued to prevent ballots from being counted if they were received after 7 p.m. on Election Day.

Context: This was one of the flurry of cases around the country seeking to limit the counting of mail-in ballots.

Status: Judge James F. Bass of the Superior Court of Chatham County dismissed the case on Nov. 5 for lack of evidence.

Wisconsin

Langenhorst v. Pecore

Filed: Nov. 12

Claim: Four voters sought to exclude all of the votes cast in three counties from Wisconsin’s total based on differences in absentee voting rules among the counties. The suit objected to Milwaukee, Dane and Menominee Counties allowing voters who say they are “indefinitely confined” by age, illness or disability to cast ballots without proving photographic identification. The complaint also cited a handful of voters (identified by only their initials) who said they received absentee ballots without requesting them, and three absentee ballots allegedly completed after they were mailed to deceased people.

Context: This suit makes the same kinds of equal-protection claims as the ones in Brooks v. Mahoney. Both were filed by the conservative lawyer James Bopp.

Status: The case is pending in U.S. District Court for the Eastern District of Wisconsin, Green Bay Division.

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