Deliberate mistake: At the start of the episode the Boy Scout asks why the Cossacks are in Gotham City, then a moment later as narrator William Dozier says, "At the Gotham City Museum" the closeup we see shows "Los Angeles County Historical and Art Museum" over the museum doors.
Catwoman's Dressed to Kill - S3-E14
Visible crew/equipment: At GCPD, when Commissioner Gordon wonders how to catch Catwoman before she wreaks even more havoc, at the start of the next shot just as Barbara stands up, the boom mic casts a shadow on the wall as it moves, on the left side of the screen. (00:05:05)
The Foggiest Notion (2) - S3-E12
Continuity mistake: When Batman finds Ye Three Bells at the docks, in the exterior shots the entry door has a handleset on the exterior side of the door, but in the interior shots when the entry door is open, we can see there's a door knob on the exterior side of the entry door. (00:06:25)
The Foggiest Notion (2) - S3-E12
Visible crew/equipment: When Batgirl enters the cricket pavilion at Ffogg Place, two chalk T-marks are visible on the floor between the snuffboxes case and the diamonds case. (00:11:35)
How to Hatch a Dinosaur (2) - S3-E9
Continuity mistake: When Barbara delivers the rare book on paleontology to Professor Dactyl, during this scene Dactyl holds the book with its spine either facing toward the right or the spine facing the opposite way, depending on the angle of the shot. (00:07:00)
Minerva, Mayhem and Millionaires - S3-E26
Visible crew/equipment: At the start, in the two consecutive scenes with millionaires William Dozier and then Howie Horwitz, while they're being treated to Minerva's renowned scalp massage, the chalk T-mark is visible on the floor at the lower left side of the screen, and in the Horwitz scene Minerva's small mark is also visible on the floor behind the spa chair. (00:00:45)
Minerva, Mayhem and Millionaires - S3-E26
Visible crew/equipment: Minerva tells Freddie that if she'll get the diamonds from the Wayne Foundation vault, he can fence them. When it cuts to Commissioner Gordon, Batgirl, and Chief O'Hara pacing the floor, three small white tape marks are visible on the rug in front of the coffee table, where each of them stop and turn on their marks. (00:05:35)
The Entrancing Dr. Cassandra - S3-E25
Visible crew/equipment: After Batman discovers Cassandra's true identity in the alchemy book, he takes a call from Commissioner Gordon on his cute portable Batphone, then he and Robin head to Spiffany's. When it cuts to Spiffany's, just as narrator William Dozier says, "Gotham City's fashionable 15th Ave," the actor's white marks are visible on the floor under the table. Then we see Cassandra's and Cabala's white T-marks when they walk in. Next, upon the Terrific Trio's entrances, we see their T-marks as well. (00:08:25)
The Entrancing Dr. Cassandra - S3-E25
Visible crew/equipment: After Alfred has placed the two-dimensional Terrific Trio into the Three-Dimensional Bat-Restorer, when they all walk out of the chamber, two chalk T-marks are visible on the floor. First we see Batgirl's T-mark and then Robin's. (00:16:50)
I'll Be a Mummy's Uncle - S3-E23
Visible crew/equipment: When King Tut runs back up the shaft and encounters Batgirl, Commissioner Gordon, and Chief O'Hara, he is reveling giddily in the truth he has discovered. Then, when Batman and Robin show up, their chalk T-marks are visible on the ground. (00:21:10)
The Great Train Robbery (2) - S3-E22
Visible crew/equipment: When Shame has finished reading Batman's skywriting message, as he says, "Darned if I won't be there," the shadow of the boom mic dips into view at the top of the green door, at the right side of the screen. (00:17:20)
Nora Clavicle and the Ladies' Crime Club - S3-E19
Visible crew/equipment: When Batman and Robin are seated in the Batmobile with Batgirl standing alongside, while they listen to the ridiculous police dispatcher, the reflection of the moving boom mic is visible on the front windshield's chrome trim. (00:10:00)
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: 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)
Louie's Lethal Lilac Time - S3-E18
Visible crew/equipment: Commissioner Gordon and Chief O'Hara think Batgirl has joined Batman and Robin inside Louie's hideout, so they decide to stay put where they are. In the next shot, when two of Louie's goons report that the Batmobile is outside, the white tape mark is visible on the floor by their feet. (00:06:15)
The Joke's on Catwoman (2) - S3-E17
Visible crew/equipment: At the start, when the crime fighting trio split up Batgirl heads to her motorcycle, and when she attempts to start it two chalk T-marks are visible on the ground, one under the motorcycle in front of the rear tire and another beside the front tire. (00:00:35)
Visible crew/equipment: When Olga shouts, "Da, da, da! Drowning her in caviar!" in the overhead shot, the boom mic repeatedly dips into view over Olga, at the top of the screen. (00:21:55)
Visible crew/equipment: After Egghead dumps Batgirl into the vat of caviar, Egghead, Olga, and the Cossacks attempt to escape, then just as they run out the door, a bit of set lighting equipment (which wasn't in previous shots) is visible at the left side of the screen. (00:22:10)
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: 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)
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