Continuity mistake: When Bugs is in Rocky's hideout, Rocky tells Mugsy to take Bugs to the other room and you can see the door is an entrance to an ordinary room. But when the police arrive later on, the door has changed to the entrance of the house.
Continuity mistake: When Mugsy falls onto Rocky after Bugs "let him have it", you can see the cigarette falls out of Rocky's mouth, but when Rocky punches Mugsy, the cigarette is back in his mouth again.
Continuity mistake: When Bugs leaves the bank after getting his carrot, you can see there is nothing on the wall next to the door. But when he says "Taxi!" in the next shot, a plaque saying "City Bank" appears from nowhere.