When Billy is shot at as he is leaving the real estate developers office the bullet shatters the back windshield and leaves a clear exit hole in the front on the passenger side. During the brief chase sequence directly afterward the the hole in the front is missing
When Billy and the mayor are in the mayor's office where he first hires Billy to investigate his wife, the decanter and the glasses on the table and the blanket on one of the couches all change position repeatedly between shots.
Exterior and interior shots show that Paul Andrews is inside Grand Central Terminal. He purchases a ticket to go to Montauk, however, there are no trains presently going from Grand Central to Montauk. Andrews then boards what is clearly a Long Island Rail Road train (a sign is outside), which does service Montauk. But then Andrews would be boarding at Penn Station, not boarding an outside station that appears to be on Long Island.
In Jack Valliant's elections headquarters there's a poster with the Hebrew word for "Jewish". Though those are the correct letters of the word, they're written left to right instead of right to left (hsiwej).