Revealing mistake: When Bruce Wayne is heading to the hearing at the courthouse, he checks his revolver to make sure it's loaded. His intent is to execute Joe Chill for killing his parents, so he would need a working gun. But when he opens the cylinder of the revolver to make sure it's loaded, you can see that all of the bullets have had the primer removed (the end of each bullet casing has a hole in it.) This is done to make each round impossible to fire, for safety on the set.
Revealing mistake: When Batman flees from Gordon after their first meeting, he leaps across an alley and catches himself on the railing on a fire escape. This fire escape fails, dropping him (as well as the clothing left to dry on the railing). Batman grabs the next railing and again, this second fire escape collapses. Exactly the same as the first, with the same clothing falling off. When he climbs onto the 'third' fire escape, we see he's only come down one level, not two. The scene was artificially lengthened by using two perspectives of the same stunt. This isn't the same as showing one explosion from multiple angles because it is timed/edited precisely to appear as a longer fall in continuous time, not as repetition of the same moment.
Revealing mistake: When Alfred is putting Rachel in the back seat of his car to drive her home, he fails to support her head. In the shot from inside the car, as he lowers her, Rachel instinctively and consciously stops her head at about 45 degrees and keeps it there until the shot is over. Had she really been unconscious her extremities would be limp and the momentum of her body being lowered would whip her head back, possibly injuring her neck. In the very next shot, Rachel is seen from Alfred's perspective, her body is already on the seat with her head only falling onto the seat from a couple of inches.