Plot hole: When The Penguin is controlling the Batmobile, Batman punches through the floor to take off the transmitter. We still see a video feed of The Penguin. Why? Batman pulled off the transmitter, so there is no reason for there to still be a video feed.
Revealing mistake: In one scene, the Penguin is making a "bat" shadow puppet on the ceiling. But there's no light behind his hands. If there's no light, there's no shadow. (01:03:45)
Continuity mistake: In Batman's first battle with the Red Triangle Circus gang he manages to escape from a fire eater by turning his batmobile around and burning the guy with the rear exhaust jet. He does this by flicking a switch (outlined in caution symbols) next to the steering wheel. When the Penguin takes Batman for a ride there is a television screen where the switch is supposed to be. When Batman blows the Batmobile apart to get through a narrow alley the switch is back in place. Not only that but the switch seems to control just about every major system on the car including the lights, shield cover, gun turrets (in the first movie), canopy etc.
Continuity mistake: When Selina returns home from work she takes milk out of the fridge and pours her cat some, leaving the carton on the worktop. She then runs to work, and when she returns after being changed to 'catwoman', the milk has vanished from the worktop. The dishes also move and plant pots appear on the window sill. (00:22:50 - 00:28:45)
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them. (01:20:50)