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 lead to their cloaked backs. (00:21:40)
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)
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)
Hot Off the Griddle (1) - S2-E3
Revealing mistake: During the fight in the museum, one of Catwoman's henchmen slams into the wall beside a window. The set wall buckles and caves in, leaving the bottom edge of the window bent and jutting out. A few shots later, though, both wall and window have repaired themselves. (00:14:00)
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)
The Greatest Mother of Them All (1) - S2-E9
Revealing mistake: When Ma Parker activates her jet-propelled wheelchair, the much younger stunt woman's face is visible as the chair speeds down the hallway toward Batman and Robin. (00:09:15)
An Egg Grows in Gotham (1) - S2-E13
Revealing mistake: A lot of the time, especially in close ups, you can see the line between the bald dome wig and Vincent Price's real skin, as well as the different coloring of skin.
The Joker's Provokers (2) - S2-E22
Revealing mistake: When Alfred talks to his twin Egbert, as they face each other, you can plainly see the line dividing the screen so that Alan Napier can play both parts.
Revealing mistake: After Shame and his baddies steal Bruce's limo. Bruce shows Dick and Alfred that he had the remote control unit keep the Bat-cycle (with Alf-cycle) following them at a safe distance. In the shots facing the Bat-cycle, attached to the front wheel there's a rope pulling it. (00:17:05)
The Penguin's Nest (1) - S2-E27
Revealing mistake: When Bruce, Dick and Aunt Harriet get into what's supposed to be an elevator, there's solid concrete under the sliding doors where there should be a break separating the lift car from the building's floor. It's visible again in Part 2, when Batman and Robin use the same elevator. (00:04:30)
The Puzzles Are Coming (1) - S2-E31
Revealing mistake: While The Puzzler and his baddies are at Artemus Knab's penthouse, across the street from Commissioner Gordon's office, when Puzzler decides he will launch the plane himself, the wire attached to the top of the plane is visible. (00:01:15)
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 Zodiac Crimes (1) - S2-E37
Revealing mistake: When Joker enters Commissioner Gordon's office at GCPD, he's pointing a wand at Gordon and Chief O'Hara. In the shot facing Joker when he lets go of the wand, the wires at both ends of the wand are visible. (00:01:15)
The Zodiac Crimes (1) - S2-E37
Revealing mistake: After Joker's goons kidnap Crustash, the opera singer, Penguin tries to sneak off, and when Batman and Robin use javelins to capture Penguin, the glistening stunt wire attached to the javelin is visible in the wide shot. (00:18:15)
The Joker's Hard Times (2) - S2-E38
Revealing mistake: When Joker pays another visit to Commissioner Gordon and Chief O'Hara, Joker's holding his wand, and in the shots from outside the GCPD office window we can see the wire cable dragging behind Joker, which is attached to the wand and providing its power. During these shots, the actors' small white tape markers are also visible on the rug, beside the Batphone table. (00:03:00)
The Penguin Declines (3) - S2-E39
Revealing mistake: At the Batcave, when Penguin is up on the atomic pile scaffolding Batman grabs hold of him just as Pengy uses his umbrella to float down, and the stunt wire attached to the umbrella ferrule is visible while the duo descend together. (00:21:05)
A Riddling Controversy (2) - S2-E46
Revealing mistake: When Batman and Robin use their jets to get out of the quicksand, as they fly up, you can see the wires used to pull them up.
The Joker's Last Laugh (1) - S2-E47
Revealing mistake: The counterfeit 100 dollar bills that Gordon and O'Hara say look "so authentic" don't actually resemble any known U.S. currency at all. In fact, a close look reveals them to be very bad photocopies of Mexican 100 peso notes. (00:01:00)
Catwoman Goes to College (1) - S2-E49
Revealing mistake: The crowd scenes during what is supposedly a Gotham City University student protest rally are inserts of rather blurry old stock footage. Not only is the mob far too large for a mere college rally, the footage is in black & white, while the rest of Batman is in color. Bit of a dead giveaway. (00:23:00)
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