(at around 34 mins) When Mike Sullivan walks through the nightclub to get information from the owner, the club members are wearing "Flappers" and "Sheiks." Both were very popular clothing styles from 1925 to 1928. In 1931, they were considered outdated.
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.
(at around 28 mins) When Mr. Rooney comes over to the Sullivan driveway in the morning and gives young Michael a quarter as a dice payoff (and for his silence for what he saw the previous night), the quarter is a Washington head quarter. The year in the movie is 1931 and the Washington head quarter started first minting in 1932.
(at around 47 mins) When Michael and his son first arrive in Chicago, a brief shot shows their car traveling into town. In the Chicago skyline, half of a (skyscraper) construction sky crane can be seen at a distance.
The heavy rain storm that occurs that fateful night doesn't appear to wash away the snow. The snow outside the Sullivan's house can be seen the next morning to be the same as it was before, even with well-worn tyre tracks in it!
(at around 1h 30 mins) When Michael, Sr. is assembling the Thompson submachine gun, in the overhead view, the magazine is on the table to the right of the case. In the next cut, the magazine is still on the table but positioned closer to the lamp. In the next scene, after Michael, Jr. walks in, the magazine is missing; it's no longer on the table, and as Michael, Sr. turns, we see that it's not mounted on the gun.
(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.
(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.
In Rance's hotel room Rance is presumably naked under a bathrobe and he makes a big spectacle of sitting down and crossing his bare legs and feet when he calls room service for a boiled egg, but when Sullivan enters the room shortly thereafter (as Rance is eating his egg) Rance is still unclothed in the bathrobe, but now has on black socks and dress shoes but his legs are still bare.
(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 un-deformed, which would be impossible.
When Sullivan assembles his 1921 Colt Thompson the last part he slides into place with a clunk is the pistol foregrip. But on real Tommy Guns that part does not slide on, but is held onto a support bar with a single long screw.
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 Maguire shoots at the car outside the diner, he is shown only a few meters away from the car. This would be impossible since the Sullivans drive off while Maguire is still seating inside the diner.
(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.
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 goof items below may give away important plot points.
(at around 1h 22 mins) During the shoot-out in Mr. Rance's hotel suite, Sullivan shoots at Maguire from behind the empty trunk. The reports heard from his .45 pistol are out of sync with the shots he fires.
When Michael guns down John Rooney he clearly has not shaven for a number of days. However when he kills Connor Rooney on the same night he is clean shaven, but then when Michael returns to Michael Jnr the stubble appears again.
(at around 1h 30 mins) When Michael is recovering from his gunshot wound, he is flipping through the papers taken from the accountant and sees multiple receipts made out to Connor Rooney. The word "receipt" on every one of them is misspelled.
(at around 1h 40 mins) When Michael Sullivan shoots Connor Rooney in the bathtub, he fires his .45 pistol three times, and three shell casings hit the floor. When he fired the shots, the hammer is never cocked and the slide of the pistol never retracts, so the chamber never actually opened to eject the spent bullet shells.
When Michael Sullivan is in the car with Michael Jr. before he robs a bank, he puts the pistol slide forward, then pulls the slide back again. By doing that, he would have sent the first round out of the chamber.