Continuity mistake: When the Penguin hooks the umbrella on Cat Woman he turns to our left to watch her fly away. A shot later he is facing straight, but a shot later he has suddenly moved 2 meters to the right and is facing the right side.
Continuity mistake: After The Penguin dies Batman's car drives next to a white car with its windshield covered in snow. From the shot inside Batman's car the snow is gone.
Continuity mistake: When the clown lies unconscious there's a string several centimeters behind his head. A shot later his head is on top of the string.
Continuity mistake: When Selina is about to taser the clown she puts on an evil face, but a shot later from a different shot she is staring with a calm look.
Continuity mistake: When Selina watches the clown's hand holding the taser, his fingers are clinched around it. A shot later his index finger is sticking out.
Deliberate mistake: After Batman destroys The Penguin's controlling system and breaks the TV screen he is 4 meters away from a narrow alley. Shot changes and he's now downtown driving around the city, Then there's a POV shot of him driving towards the narrow alley again, though he is now many meters behind. Then back to shots of him driving downtown until he finally arrives at the narrow alley. It adds up for pace, but it's very awkward indeed.
Continuity mistake: When Catwoman exits Schreck's she's doing flips without hells. After the explosion, when she climbs the wall heels appear.
Continuity mistake: When Batman talks with The Penguin, before Catwoman arrives, The Penguin's left arm is raised or not depending on the shot.
Continuity mistake: Catwoman turns the microwave dial to the right, pointing in a 3 o'clock position. A shot later it points in a 12 O'Clock position. (00:56:04)
Continuity mistake: Inside the cage, Max covers his mouth with his hand and then uncovers it and grabs an iron bar. A shot later he is suddenly covering his mouth again. (01:37:16)
Revealing mistake: When The Penguin controls the Batmobile, before it crashes against a car on the left and sends it flying away, check the car's rear wheels: they're not touching the road.
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.
