The Puzzles Are Coming (1) - S2-E31
Revealing mistake: At the old balloon factory, when Puzzler throws the model airplane with a poison tip at Batman, on his upper arm we can see the protruding outline of the rectangular base to which the small airplane is attached. (00:22:00)
The Puzzles Are Coming (1) - S2-E31
Visible crew/equipment: At the Knab Aircraft hangar Puzzler arrives in the truck just as Knab is leaving, and when Puzzler walks to the back of the truck the reflection of the boom mic is visible on the right back door windows. (00:14:45)
The Joker's Provokers (2) - S2-E22
Continuity mistake: While Joker and his baddies are in the vent shaft at the waterworks, Joker takes out the envelope and holds the envelope with his left palm up, but in the closeup his left palm is down as he pours the pills into his right hand. (00:14:50)
Tut's Case is Shut (2) - S2-E8
Revealing mistake: While Chief O'Hara is outside on the ledge of the top floor, King Tut commands him to use the flagpole to do acrobatics. When O'Hara obeys Tut, the vertical wire at the end of the flagpole is visible. (00:08:20)
Walk the Straight and Narrow (2) - S2-E2
Visible crew/equipment: After Archer and his goons have hit the armored truck, when the security guard pulls out the bag of money from the back of the truck and shows it to his partner, the chalk T-mark is visible on the ground between their feet. (00:10:20)
Walk the Straight and Narrow (2) - S2-E2
Revealing mistake: When the armored truck with the ten million is hit, during this scene we can see stunt wires for some of the arrows used by Archer and his goons. (00:09:00)
Walk the Straight and Narrow (2) - S2-E2
Revealing mistake: At the start, while Archer and his goons are riding toward Batman and Robin who are tied to the stakes, when the Dynamic Duo use their Bat-springs the stunt wires lifting them are visible. (00:03:05)
Shoot a Crooked Arrow (1) - S2-E1
Revealing mistake: In Commissioner Gordon's office, just as Gordon tells Dale that with Batman and Robin on their side no harm will befall the money, the top of the set is visible at the top right corner of the screen. (00:06:25)
Shoot a Crooked Arrow (1) - S2-E1
Revealing mistake: At Archer's underground hideout, Batman and Robin are strung up by their thumbs, but in fact the stunt wires that lead to their cloaked backs are visible. (00:21:40)
Fine Finny Fiends (1) - S1-E33
Visible crew/equipment: When Batman and Robin enter Penguin's hideout at the South Pier, with Penguin and his goons ready and waiting for them, the duo walk to the table and both of their chalk marks (a bit smudged) are visible on the floor where each stands. Then, when Penguin captures Robin and places him against the wall, another two chalk marks are visible on the floor for each actor. (00:20:40 - 00:22:05)
While Gotham City Burns (2) - S1-E30
Continuity mistake: While Bookworm is at stately Wayne Manor he hands Aunt Harriet a book, and when she opens it Harriet and Alfred are gassed unconscious. In the next closeup of the rare book encased in glass, its open pages have colorful illustrations, but in the next shot the open pages have only plain text. (00:09:30)
The Bookworm Turns (1) - S1-E29
Visible crew/equipment: When Printer's Devil contacts Bookworm headquarters, just as he messages "plot A washout" Bookworm walks over to the desk, and the white tape mark is visible on the rug where Lydia will stand. (00:07:25)
The Joker Trumps an Ace (1) - S1-E25
Revealing mistake: In Commissioner Gordon's office, when Batman opens Joker's gift the genie figure begins to inflate, and in the overhead shot the stunt's covered air hose is visible at the bottom of the screen. In the next shot Batman pops the figure causing it to deflate, and the wire at the top of the figure's head is visible as it descends. (00:07:05)
Revealing mistake: While Batman and Robin are tied up and suspended over the enormous candle dipper vat, there are stunt wires leading to under their capes. (00:21:00)
Better Luck Next Time (2) - S1-E20
Revealing mistake: When Batman dodges the tiger by using metal claws to climb the wall, the stunt wires lifting him are visible. (00:03:00)
Better Luck Next Time (2) - S1-E20
Continuity mistake: While Batman is increasing his communicator's audio modulation so as to incapacitate the tiger, Batman is wearing Bat Ear Plugs to protect his hearing. As Batman turns his head in the closeup, we're able to see that the Bat Ear Plug has vanished from his right ear, then it reappears in its proper place. (00:04:55)
The Purr-Fect Crime (1) - S1-E19
Visible crew/equipment: When Commissioner Gordon uses the Bat Phone and Alfred answers, it then cuts to Bruce and Dick playing crazy chess, and as the camera moves for its dolly shot, two of the crew's wood planks with marks become visible on the floor. (00:02:15)
Deliberate mistake: When Batman and Robin get free from the train tracks in the nick of time, the shot of the couple on the passing train is flipped; note things are backwards. (00:04:30)
Batman Stands Pat (2) - S1-E14
Visible crew/equipment: Batman and Robin have been captured so they're led to Mad Hatter's hat factory. When Lisa arrives, a chalk mark can be seen where she stands, and a white tape mark on the floor for Mad Hatter. (00:19:30)
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)
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