The bullet hole in the rear window of the Sullivan's car disappears and reappears.
(at around 9 mins) In the first scene with Connor, the first shot shows his cigarette with a long build-up of ash on the end. In the very next shot, however, the cigarette has regenerated to near-full length, with very little ash.
Maguire's coffee cup magically appears between shots.
(at around 42 mins) Michael Sr. tells Michael Jr. that this is no longer their home, then walks away from his son to the car. But when he gets there, his son is already in the car.
(at around 10 mins) When Mr. Rooney speaks to the crowd at the wake, he initially stands on a chair. In the next shot, he stands beside the chair, which is sideways to the crowd. In the next shot, the chair is facing the crowd.
Bullets fired into Michael's back at the lake house exit his chest with enough velocity to cause blood spatter on the picture window inches in front of him but don't have enough energy to cause any damage to the window itself.
When the Jude Law character fires at Sullivan's car fleeing the diner, two bullets penetrate the car's rear window and disappeared, rather than hitting someone or continuing on to penetrate the windshield.
In the end, after Maguire focuses the camera he then snaps the picture without ever having loaded a film holder. One could believe that he has already cocked the shutter. But there was not enough time elapsed to go through the film loading procedure.
When Michael (Hanks) guns down Finn's henchmen in the warehouse, the spent casings from the Thompson machine gun he fires drop between his legs when they should have been ejected to his left at least a couple of feet which would be out of the camera frame which is shot from Michael Jr.'s POV.
During the gunfight in Rance's room, the shot that shatters the cut glass lamp blinding the assassin is fired from Sullivan's 1911 when the gun was empty as revealed by the slide being locked rearward.
(at around 1h 4 mins) In the diner scene, a bottle of HP Sauce is on Sullivan's table. HP Sauce is available almost exclusively in the United Kingdom; it wouldn't have been in an average American roadside diner in 1931. Director Sam Mendes used to be the artistic director at the Donmar Warehouse in London and developed a fondness for HP Sauce. He deliberately had that bottle placed there.
(at around 1h 24 mins) The bullet removed from Sullivan's arm has no rifling marks, and appears to be .45 caliber, rather than the .32 that Maguire was using. It's also completely undeformed, which would be impossible.
(at around 1h 8 mins) When Sullivan and his son drive off the road after the diner scene, tire tracks are already there from previous takes.
As stated in the beginning and ending statements by "Michael" the entire movie (story) takes place in the "winter of 1931". However, at about the 56-minute mark Maguire is shown walking under elevated train tracks. The leaves on the trees to the right side of the screen under the elevated train tracks are green and fully leafed. The scene outside the diner includes the sound of crickets chirping and shows long, green grass going to seed alongside the road. So from the 56-minute mark to the end credits (with one brief exception, a shot outside the church when John Rooney is taking communion) the remainder of the movie was obviously filmed in either very late spring or full summer with all of the trees being green and fully leafed and all of the grass being green and tall.
The bank records Sullivan examines have the word "receipt" misspelled as "recipit."
When Michael guns down Rooney's men and Rooney in the heavy rain, no steam is shown coming off from the barrel of the machine gun. Shooting so many rounds in a short period of time would have made the barrel red hot.
In that era, gentlemen removed their hats indoors, particularly in places like diners. Even not-so gentlemen. To not do so would have attracted attention.
Michael Jr. reads "The Lone Ranger" in 1931. It was created for radio in 1933; the books were published even later.
(at around 30 mins) Parts of the American flag are visible in Michael Jr.'s school. From the spacing of the visible stars, the flag clearly has 50 stars, not the 48 it would have had in the 1930s.
At the wake, the men are seen smoking filtered cigarettes. Though this is set in the 1930s, the first filtered cigarettes would not be introduced until 1954 by the RJ Reynolds Tobacco Co.
When Michael and his son first arrive in Chicago, there is a brief shot of bright silver metal cars on the El. They were introduced in the 1980s. In 1931, the cars would have been made of wood and painted dark brown.
Sullivan knew that Maguire was chasing them and that he knew about the beach house after the diner incident. But after killing Connor, he takes his son there anyways without giving a thought that Maguire is still alive, could still be chasing them, and knows they would have headed there.
When Sullivan ambushes Rooney, his bodyguards do return fire, but they do nothing to protect him. Surely they would have tried to shield him or pull him away from the gunfire.