Continuity mistake: When Batman and Robin scale the warehouse wall then watch Joker and his cohorts, each of the two windows have latches at the bottom, but when the Dynamic Duo burst in through the window, only the left window has a latch. (00:11:45)
Visible crew/equipment: After Joker reveals to his cohorts that his plan involves champagne, when it cuts to Commissioner Gordon standing with Batman at the microphone, as the camera zooms out the chalk T-mark is visible on the floor in front of Batman and another actor's mark in front of Chief O'Hara. (00:18:15)
Visible crew/equipment: When the museum closes early and Joker calls to each of his cohorts, their individual chalk T-marks are visible on the floor where they all stand. (00:12:40)
Visible crew/equipment: When both Batman and Robin realise that even though the museum can't be broken into, it doesn't mean someone couldn't break "out" of the museum, as they rush to the museum's front doors we can see a chalk T-mark on the ground in front of the doors. (00:13:25)
Visible crew/equipment: At the end, Joker announces the unmasking of Batman and Robin, then when Joker turns and walks toward them, multiple chalk T-marks are visible on the floor where the dynamic duo are being held. (00:23:55)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: When Penguin uses the large umbrella on his penguin line to slide across to Dawn's penthouse terrace, as he says, "Who says penguins can't fly," we can see the glistening of the crew's wire at the top of the pulley wheel, as it moves. (00:13:45)
The Penguin's a Jinx (2) - S1-E4
Visible crew/equipment: When Penguin is riding the large umbrella across the Penguin line in the wide shot, just as he is about to reach Dawn's terrace, we can see a crew member's two arms reaching out to Penguin and the top of a ladder, at the lower left side of the screen. (00:13:50)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: When Penguin uses the large umbrella to slide across the Penguin line, at the top of the umbrella there's a pulley wheel, but when he reaches Dawn's penthouse terrace there's only a small loop at the top of the large umbrella. The same thing happens when Swoop follows Penguin on another large umbrella. (00:13:40)
Fine Feathered Finks (1) - S1-E3
Revealing mistake: At the start, Penguin's henchmen give out booby trapped umbrellas to everyone outside the House of Ali Baba. When they all enter the jewelry store those umbrellas explode with fireworks, gas, confetti, etc, and the stunt wires holding up the umbrellas are visible, at the top of the screen. (00:01:20)
Fine Feathered Finks (1) - S1-E3
Revealing mistake: After the huge rooftop umbrella is launched onto the street there's a small umbrella hanging from its handle, and when Batman uses his Batarang to swing towards the huge umbrella's shaft, his stunt wires are visible. (00:18:45)
Fine Feathered Finks (1) - S1-E3
Visible crew/equipment: In Commissioner Gordon's office, when Gordon is looking over the paperwork of three new umbrella factories, frustrated that none are in the name of Penguin, as he stands up we see two white tape marks on the rug behind him. In the previous scene with Warden Crichton, four of the actors had been standing in that area. (00:11:35)
Continuity mistake: When Robin has the idea that Riddler placed two other riddles hidden in the legal documents, Robin holds the papers up with his raised right hand, but in the next shot the documents are being held up in Robin's left hand. (00:13:45)
Continuity mistake: At the secret HQ of the Mole Hill Mob, Molly is wearing a ring on her right hand and many bangle bracelets as she dances, but in the closeup when reaches into the caviar jar the ring and bracelets are gone, then it all reappears. (00:15:00)
Continuity mistake: During the fight, the desk mat and desk calendar are moved initially and then moved again later in the fight. (00:19:03 - 00:22:00)
Revealing mistake: At the start, when Joker has sprung himself from prison, there are three shots of the giant spring, and the wires holding up the platform at the top are visible. In the third shot, one of the wires even glistens for a moment as it moves. (00:02:35)
The Bird's Last Jest (2) - S2-E28
Continuity mistake: When the Dynamic Duo goes up the Bat Pole chute, the NEGATE COSTUME CHANGE lever is down. When they come down as Dick and Bruce, the lever is up, and Robin pulls it down to stop the instant change back to Dick and Bruce.
The Clock King's Crazy Crimes (1) - S2-E11
Continuity mistake: After starting the sand in the giant hourglass, Clock King puts on his alarm clock top hat. Between distance and close shots, the clock reads a few minutes to three, then Three o'clock in the closeups.
It's the Way You Play the Game (2) - S2-E26
Visible crew/equipment: After Batman gets out of the Batmobile, you can see the lighting rigs, boom mic, camera and crew reflected on the Batmobile's side.
Death in Slow Motion (1) - S1-E31
Audio problem: In the distance shot, when Robin responds to Batman's call, his lips don't move, but you hear him.
Not Yet, He Ain't (2) - S1-E22
Continuity mistake: When the Batmobile is blocked by the second police car, it's stopped on an angle, and almost touching the car. When the Penguin commandeers the Batmobile, it's sitting correctly on the road, so that it can be driven easily without really needing to back up.
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