Continuity mistake: When Penguin removes the manhole cover above him right before the Mayor's baby is stolen, he places it right next to the edge of the manhole, but when the baby snatcher jumps in, it is further away.
Audio problem: When Selina returns home from work the first time and Miss Kitty comes in through the window, Selina pours some milk into the cat bowl and Miss Kitty begins to lap the milk (out of shot). Then, without a break in the lapping, Miss Kitty meows again as a cue for one of Selina's lines, but the lapping sounds continue underneath the meow. It's an open-mouth meow, too, and, as far as I know, no cat can both meow and drink at the same time. (00:22:45)
Continuity mistake: At Wayne Manor, Bruce receives the projected bat signal while in his study. As he stands up, the circle of light hits only the last shelf on the bookshelf behind Bruce. But in the next shot, the light circle is much lower and hits almost the entire bookshelf. (00:13:10)
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.
Audio problem: When the mayor's baby is first stolen, the audience gasps in a very fake way. When the Penguin reveals his plan to steal the first-born sons to the party members later in the movie, they gasp the exact same sound effect. They actually do it a third time, later in Penguin's speech, but it's very quiet.