Continuity mistake: Just after Bruce has spoken to Alfred about him letting Vicky Vale into the Batcave in the past, we see him pulling his jacket on from behind. He puts his right arm in, then starts to reach behind himself to get his left arm in. The shot then cuts to seeing him from the front, and we see him reach behind himself again.
Continuity mistake: When the circus and Batman first make their appearance, one of the circus men grabs Selina Kyle and threatens Batman with a Stun Gun. After the man is knocked unconscious, Selina "zaps" the man. I don't know how many of your are familiar with stun guns but when they are activated on someone or something they don't spark or make a sound. The sound is only heard if held up in the air and is not in contact with anything. Yet when she zaps the man, there is a clear shocking sound.
Continuity mistake: When there's an overhead shot of the crowd in front of the Hall of Records, there is a person in bright red clothing amidst the people near the doors, pushing against a police officer, but in the next shot when Shreck is speaking to the press, the person is nowhere to be found and could not have exited the crowd in that short time.
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.