Continuity mistake: Just before Batman takes his mask off at the end, the black makeup around his eyes vanishes while the mask is still on.
Continuity mistake: In Schreck's department store, when Catwoman is doing her cartwheels, her heels collapse.
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.
Deliberate mistake: When the penguin goes into the cemetery, he bumps a tombstone and it moves. Almost certainly a tribute to legendarily poor director Ed Wood, who Tim Burton later directed a biopic of - this exact thing happened in one of his films.
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.
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them.
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.