Plot hole: Right after the Joker goes into the cathedral with Vicky, Batman follows him with no one else entering in between the two parties. How then did Joker's goons get to the top of the cathedral before Batman or the chopper? They couldn't have planned it since Joker had to tell them where to pick him up, and they didn't know he would chase after Vicky to that particular location.
Revealing mistake: Right after Joker falls off the top of the cathedral, Batman and Vicky fall too and we catch them three times before Batman's grapple digs into the stone. On the second and third time you can actually notice Vicky's face expression doesn't look one bit scared, so you know they're not falling. You can also see there's a screen in the background. (01:58:40)BigOLB
Continuity mistake: When Batman jumps off of the remaining bell at the top of the cathedral, the goon uses Batman's momentum to throw him into a set of stairs, his trajectory parallel with the near wall. Since the goon wasn't standing near the wall when he threw Batman, this angle would not be possible to achieve.
Visible crew/equipment: When Batman comes through the skylights in the museum just after Vicki throws water on Joker's face, you can see the stunt cable that lowers Batman down. Batman does use a system of cables and grappling hooks because he has no superpowers, but the cable lowering him down disappears as soon as he hits the ground. It was used to lower the actor down safely, not the character.THGhost
Continuity mistake: In the scene where Joker has broken in on the mayor's televised speech to the citizens of Gotham to challenge Batman to a "big duke-a-roo", Bruce freezes the live broadcast on a close-up of the Joker. He then flashes back to his parents murder. After the flashback, he looks again at the freeze frame picture of the Joker, but it is a different picture. When he freezes it, Joker's eyes are closed. When he comes back to the picture, his eyes are open.
Continuity mistake: At the cathedral scene, when Batman & Vicky are hanging at the ridge, and the Joker is about to be picked up by the helicopter, the color of Joker's jacket changes from purple (front shot) to red/orange (shot from above) and back again. The orange one may be his blouse, as it has the same colour visible under his jacket in the front shot. Then, the same jacket and hat are seen hung over the gargoyle just a few seconds later.
Revealing mistake: When Batman drives the Batmobile into the Axis Chemicals factory, watch closely when he slams through the first gate - you can see the gate is swaying and starting to open before the batmobile hits it, indicating it wasn't really closed and was left unlocked and partially open to make it easier for the stunt-driver to drive the Batmobile through it.Dr Wilson