Continuity mistake: When Bruce goes to gear up to rescue the Ice Princess, there are visible footprints on the drawbridge that leads into the chamber where all the bat suits are kept. When he then steps back out in his Batman attire, the drawbridge is now clean of all prints. (00:00:24)
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. (01:48:20)
Visible crew/equipment: After Batman and Catwoman's mistletoe banter, a wire can be seen attached to his back as he glides off the rooftop. (01:20:15)
Continuity mistake: In Schreck's department store, when Catwoman is doing her cartwheels, her heels collapse. (00:53:40)
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.
Suggested correction: We see the Red Triangle gang spends a while fiddling with the Batmobile's workings before installing the transmitter. It's likely they made multiple 'modifications', thus the transmitter was for controlling the car's engine and steering, and the video feed was made possible by another, separate means.
Suggested correction: I concur with the other correction. There are multiple shots showing the gang working on the car and doing other things to it beyond just putting on the transmitter. (You see them playing with wires, moving parts around, etc.) Hacking into the computer/video-feed so Oswald could taunt Batman is likely among the other things they are doing. There's nothing into the movie that suggests taking off the transmitter should (or even could) interrupt the video-feed. The fact that Batman has to punch the screen to get rid of the Penguin's image is another point to the fact that the transmitter itself had nothing to do with the 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)
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. (00:39:15)
Continuity mistake: When the Penguin is looking out of the barred sewer hole, from an inside shot we see the penguin holding two bars, and in between the bars he is holding are two bars. In the next shot, there are four bars in between. (00:06:05)
Continuity mistake: Catwoman's lipstick smears down onto her chin when she puts The Penguin's bird in her mouth, yet in the shot just before she gives herself a "bath", the lipstick momentarily is gone. (01:03:20)
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)
Continuity mistake: When Selina retrieves her papers from the floor, she places them in a disorderly pile beside the clown. In the shot when she zaps the clown with the taser, the papers have changed position and are stacked neatly. (00:15:55)
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them. (01:20:50)
Continuity mistake: When Selina puts her soft toys into the sink, and begins to push them down into the shredder hole, between shots the positions of the teddies in the sink differ. (00:29:45)
Continuity mistake: When the Penguin bites the guy on the nose, while he is talking to Max Shreck afterward the blood on his face is there, then it disappears, then it's there again. (00:49:40)
Revealing mistake: Outline of the broken piece Batman will yank out of the wall to save Selina is visible before he makes it. (00:15:15)
Continuity mistake: The rubble and debris from the piece of wall which Batman tore from the wall, which lies beside the clown, differs between shots, and so does the clowns position. (00:15:50)
Continuity mistake: When Selina becomes enraged and trashes her apartment, she grabs a frying pan off of the oven hob, which was not there previously, when she entered her apartment. (00:28:40)
Revealing mistake: When The Penguin shoots the clown who says "Killing sleeping children...", as the clown falls back to the floor, you can clearly see he falls on a wrinkled, soft mat which has unsuccessfully tried to be disguised. (01:29:30)
Visible crew/equipment: When the large present drives out into the middle of the Christmas procession, as the lid of the gift springs up, a long white wire can be see attached to the lid, which wasn't there beforehand. (00:11:30)
