Plot hole: A vital plot line, obviously, is that Doyle wants to kill the con men who fleeced his runner of the numbers money. He has Luther killed and turns his best men (and women) onto Johnny Hooker, almost killing him, too. What about the third conman, Kid Erie? He is an essential part of the con, as much a part of it as Luther and Hooker. During the setup - just before they fleece him - the numbers runner watches Kid Erie running away. He looks at him, Hooker and Luther in turn. Even if he couldn't identify him he would still be able to inform Doyle that there were three rather than two con men involved. Even so, Kid Erie comes and goes as he pleases. Doyle doesn't have anyone looking for him; he doesn't even mention him in conversation, and in fact consistently refers to two - not three - con men. He makes it clear that he would have to kill his best friend if he even found out about the con, yet he lets one of the central participants go scot free. It doesn't make any sense at all.
Continuity mistake: When Lt. Snyder breaks the phone booth window with his gun and points it at Hooker, Hooker's hat falls off. Yet in the next shot over Lt. Snyder's shoulder you can see that Hooker's hat is back on his head.
Continuity mistake: When Hooker is taking Lonnegan to meet his Western Union tipster, he gives Lonnegan's driver an address. When the car pulls up to the destination, the number on the building is different.
Revealing mistake: After Hooker escapes the attack by Lt. Snyder in the phone booth, he runs down an alley. On a cross street in the distance, a modern sedan and large white truck drive by.
Continuity mistake: When Hooker meets Gondorff for the first time, he puts him in a shower, fully dressed. When the camera cuts to a front view of Gondorff, as the shot goes back and forth between him and Hooker, Gondorff's right pant leg alternates between being soaking wet and completely dry several times.
Factual error: When Lonnegan brings the briefcase full of money to the cage for his big bet, all of the 100s in the case are series 1966 federal reserve notes, which obviously did not exist in the 1920s or 30s.
Character mistake: At Duke Boudreau's bar, the potential recruits for the sting are told, "Gondorff is setting up a con on the North side". Later in the film, Agent Polk tells Lt. Snyder, "Gondorff is running a con on the South side". The actual address for the betting hall is not given, but it is just down the alley from Klein's, which does have a South side address.
Continuity mistake: When Lt. Snyder meets with Polk, the FBI agent, they both sit down and Snyder makes a comment about Polk's "nice office," which is really an empty factory. Polk gets offended immediately and from an over-the-shoulder shot you can see him rise from his chair. The camera switches to a shot of his face and he's calmly sitting down again, then rises up in anger. He stands up twice.