A 2014 report in the journal Electoral Studies — roundly criticized by other researchers as methodologically flawed — suggested that registration procedures are lax enough that as many as one in 15 non-United States citizens living in the country could have mistakenly cast ballots in the 2008 election. A 2012 study by the Pew Center on the States, cited by Mr. Trump on the campaign trail, concluded that 24 million registrations were outdated or invalid, 1.8 million registrants were dead and 2.75 million were registered in more than one state.

But even the study that Mr. Trump cited pointed to inefficient administration, not fraud. In a series of tweets, David Becker, the primary author of the study, said the study found no evidence of noncitizen voter registration or voting, and no evidence of voter fraud because of out-of-date records or deceased people still on voting rolls. He said voter rolls are more accurate now than when the study was done in 2012.

And, perhaps most important, the principal fraud that Mr. Trump and most Republicans assail, and the only one that voter identification laws address — voters who intentionally misrepresent themselves at polling places — is exceedingly rare, experts say. They add that it is almost impossible to perpetrate on a scale that would affect the results of a national election. Democrats also note that with Republican domination of state governments, voting nationwide is increasingly overseen by Republicans.

No national database of voter fraud cases exists. But a study by Justin Levitt, a professor at Loyola Law School in Los Angeles who currently works in the Civil Rights Division of the Justice Department, uncovered only 31 credible claims of voter impersonation between 2000 and 2014, out of one billion ballots that were cast. An Arizona State University journalism project reviewed 2,068 allegations of election fraud between 2000 and 2012 and concluded that only 10 had involved misrepresentation.

Corrupt Officials

More common and largely unaddressed, but still rare, are fraudulent mail-in ballots and corruption among election officials.

Fraud charges are a staple of American politics, and once were frequently true. Terre Haute, Ind., was infamous for a 1914 scandal in which the mayor rigged voting machines, bought off voters, registered thousands of nonexistent voters and arrested nosy poll watchers.

But if thievery has not vanished since then, its scope has shrunk markedly: Four Troy, N.Y., officials and party workers were convicted in 2011 of creating false absentee ballots that may have swung local elections. And the next year, Indiana’s chief elections official, Secretary of State Charlie White, was convicted of six felonies involving voter fraud, including submitting a false ballot.