Revealing mistake: When Batgirl hands Batman the rope, just as he tells her to concentrate on the Indian fakir command, above Batman's head we can see the glistening wire that will be used in the stunt. Then, after Batman and Batgirl have escaped the dungeon Robin rushes up and asks how they did it, and when Batman reiterates the Indian fakir command for Robin, we can see the wire attached to the rope. (00:15:55 - 00:18:05)
Visible crew/equipment: While Batman and Batgirl are in the dungeon attempting to escape, it cuts to Alfred and Robin meeting up at the Batmobile at the same time, and the reflection of the diagonal boom pole is visible on the windshield. (00:16:15)
Revealing mistake: The strings used to operate the bees attacking Robin are visible.
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon