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)
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them. (01:20:50)
Revealing mistake: The snow that they use floats on water in almost every scene. (00:02:00)
Revealing mistake: Catwoman breaks into the department store to blow it up. She punches her fingers through a metal access panel to get to the gas lines. Look closely at the panel - the holes are already in place at the exact locations where she punches through, they're just covered with white paint or putty. (00:55:15)
Revealing mistake: In the final scene where the Penguin's corpse is dragged in the water by the other penguins, as he's immersed in, a sort of black mat floats away from the body, which was used to make the body drag easily.
Revealing mistake: During the climax, when a rocket hits the "Arctic World" sign and explodes, subsequently causing the polar-bear statue to fall, look closely. The scale of the fire and smoke (particularly the smoke, which looks similar to cigarette-smoke size and not the enormous plumes that such an explosion would create), in addition to the unnatural movements of the alleged large "statue" reveal that this effect was achieved with a rather obvious miniature set.