When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: While Bruce Wayne is in Commissioner Gordon's office, when the Batman imposter shows up, numerous white tape marks are visible on the rug, where the actors will step as the scene progresses. (00:14:20)
When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: After the Batman imposter has met Bruce Wayne in Commissioner Gordon's office, when it cuts to the imposter returning to Riddler's hideout, the chalk T-mark and another mark are visible on the floor near Mousey's feet. (00:16:25)
When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: While Robin is "fiddling with the atomic pile" up on the scaffolding in the Batcave, Commissioner Gordon calls Batman to ask if he'd been to Gordon's office earlier, and in the overhead shot of Gordon's office there are two white tape marks visible under the desk, with one at Gordon's feet. (00:17:45)
When the Rat's Away the Mice Will Play (2) - S1-E12
Revealing mistake: When Riddler and his Rats have been caught near the end, during the ensuing fight Batman throws one of the Rats, and as he goes flying across the room we can see the stunt wires. (00:21:55)
When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: The microphone dips into the shot at the end, just as Aunt Harriet is kissing Dick and Bruce good-bye. (00:23:20)
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