Enter Batgirl, Exit Penguin - S3-E1
Revealing mistake: The panel that Batgirl makes her entrance through can be plainly seen as a prop mounted on top of the set of stairs she comes down.
A Horse of Another Color (2) - S3-E5
Revealing mistake: Ethel Merman just can't keep a straight face amidst Batman's campy silliness. Here and in the previous episode, she can be seen cracking up in several of her scenes with Burgess Meredith. (00:05:00)
Surf's Up! Joker's Under! - S3-E10
Revealing mistake: While Batman and Robin are being processed into the foam surfboard blanks, they are lying on two wood workbenches. When Riptide and Wipeout begin adding the coat of resin to both foam blanks, just as Riptide announces both blanks are smoking, we can see under Robin's wood workbench the fallen wired smoking apparatus used in the stunt. (00:13:05)
Surf's Up! Joker's Under! - S3-E10
Revealing mistake: When the Dynamic Duo are locked in the board blanks, you see the jets where the smoke and flame come out.
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)
Revealing mistake: The strings used to operate the bees attacking Robin are visible.
Nora Clavicle and the Ladies' Crime Club - S3-E19
Revealing mistake: When Robin calls Chief Linseed to get her to mobilize the entire force, Linseed and the others are useless, and in the two closeups of the mice scurrying on the rug in front of the desk, a few stunt wires are visible. (00:18:50)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Revealing mistake: Not only does Batman's solo flute play two music tracks at once, it also keeps on playing after he's taken it out of his mouth. (00:20:00)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Revealing mistake: When the crime fighting Trio are playing the tune on their toy flutes and reach the docks, in the wide shot the stunt apparatus (a bar) is visible, and in the closeups of the mice procession their wires are visible as well. (00:20:50)
Penguin's Clean Sweep - S3-E20
Revealing mistake: In any of the close up shots of Penguin, his nose has a completely different skin colour than the rest of his face. Making it very obvious that the actor is wearing a fake nose.
The Great Train Robbery (2) - S3-E22
Revealing mistake: At the Batcave, when Batman programs the skywriter drone and switches it on, the two stunt wires are visible as it heads out of the Batcave. (00:15:40)
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