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.
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)
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)
Continuity mistake: At the very end of the episode, when Batman, Robin, and Batgirl find out Joker is released from prison, one shot shows Joker in his usual pink suit. But in the very next shot, his suit is gray.
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.
The Funny Feline Felonies (1) - S3-E16
Other mistake: At the start of the episode, Bruce Wayne and Warden Crichton are overseeing Joker's release from prison, but at the tail end of the last episode, Batman is shocked that Joker is released from jail. He could not be two places at once.
The Funny Feline Felonies (1) - S3-E16
Factual error: When Batgirl checks Batman's pulse, she states that his pulse is "three beats per minute." However, she only holds Batman's wrist for two seconds. There is no way she could have determined that pulse rate in so short a time. It is unlikely she could have detected even one heartbeat at that rate in two seconds.
The Funny Feline Felonies (1) - S3-E16
Continuity mistake: When Catwoman shoots the bullet into Commissioner Gordon's office, about a quarter-inch of the bullet remains exposed. However, when Batman goes to pull it from the wall, about four times as much of the bullet is now exposed.
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)
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)
Louie's Lethal Lilac Time - S3-E18
Character mistake: When Gordon and O'Hara show Barbara the slides of Batman and Robin's previous exploits, the last slide shows the Batmobile being hit with mines. Gordon says they're chasing King Tut, but it's actually from Catwoman's "Better Luck Next Time".
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: 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.
Penguin's Clean Sweep - S3-E20
Continuity mistake: After Batman has captured the last fruit fly, he goes to put the vial away in his utility belt. In the close up shot, his hands move toward the left side of the belt. The shot changes to a long view and his hands are now instantly coming up from the right side of his belt.
Visible crew/equipment: At the Gotham City Opera House, after the Terrific Trio have taken care of Shame and his men, Calamity Jan and Frontier Fanny show up and spray the heroes with fear gas. When Shame and his gang walk over to Batman, Robin, and Batgirl, a few chalk T-marks are visible on the floor where they will stand. (00:17:45)
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