Revealing mistake: Most of the first season's episodes follow a standard outline where, after the opening credits, Batman and Robin arrive in the Batcave via the Batpoles, run and enter the Batmobile and drive off, along the winding, dusty roads toward Gotham, then arrive in front of City Hall, exit the car and run up the stairs. But very quickly it becomes obvious that every bit of this is the same footage used over and over. As they get in the car, it's on a black platform which vanishes as the car starts moving. The same cloud of dust appears as the car's rear heads into the distance on the winding road, and the same crowd of people are in front of City Hall, nearly every episode.
Revealing mistake: The Batcave was a stage set with no real exit for the car. So in many episodes where a shot is looking into the cave from the darker tunnel as the Batmobile approaches and passes to our right, the rear end of the car jumps upward and tire skid marks appear on the floor as the driver slams on the brakes to avoid hitting the studio wall that is out of the shot, while the audio suggests the car is still accelerating.
Revealing mistake: As the Batmobile exits the Batcave, if you watch the exit closely, you see the shadow of the back end of the car come to a stop, meaning that it was at the wall of the set. (00:05:10)
Revealing mistake: Riddler laughing maniacally after handing Batman the lawsuit turns around and leaves, not before tipping a vase over. There is a crashing noise as if the vase shattered, but the vase is tumbling in a full 360°, undamaged. (00:11:50)
Smack in the Middle (2) - S1-E2
Revealing mistake: Of all the possible mistakes related to the Robin mask Molly supposedly wears, the one that really could have been avoided was the fact that the camera does not cut away from Burt Ward while he is in the middle of the gesture of turning around and 'ripping away the mask', but only after he's done it, so you can tell nothing came off, nothing moves in his hair, let alone the face itself. (00:10:25)
Smack in the Middle (2) - S1-E2
Revealing mistake: Batman tosses the Batarang at Riddler after bursting through the door and yelling "Surrender, you rat." Such is Adam West's physical prowess and the strength he puts in the athletic gesture, that the mere touch of his foot moves the heavy cement block lying on the floor. That or it's made of plastic (same one he picks up later in the scene tossing it at the 'bulletproof glass'). (00:14:30)
Smack in the Middle (2) - S1-E2
Revealing mistake: While Riddler is with the unconscious guests at the Moldavian exhibit, in the basement of the Moldavian Pavilion the Mole Hill gang blow up the ceiling to create an entry to the exhibit, on the floor above. They lean two tall ladders at the side of the hole and climb up. When Riddler makes for an escape Batman follows him down below, and now we can see long metal brackets bolting the ladders to the floor for safety. (00:20:20 - 00:22:00)
Smack in the Middle (2) - S1-E2
Revealing mistake: When Batman bursts out from the mammoth, one of the 'unconscious' extras changes position reacting to the practical effect. (00:20:45)
Smack in the Middle (2) - S1-E2
Revealing mistake: After Batman tosses his Batarang up to get to Molly, the lifting wire attached to him is plainly visible.
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: In the close-ups of the basket filled with umbrellas at the bank, the asbestos blanket is continuously bumping up and shaken not by the 'explosions' of the booby trapped parasols, but rather by the thin wires you can make out at the top of the frame. (00:15:10)
Fine Feathered Finks (1) - S1-E3
Revealing mistake: The trademark vertical rope climb of this episode is done on the handle of a huge umbrella. The umbrella though is at an angle in the various wide shots, and Batman is climbing a perfectly vertical handle. (00:18:30)
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
Revealing mistake: When Penguin pulls the lever, Adam West assumes the position to receive the net dropped on his head a couple of seconds early and as if Bruce/Batman knew beforehand that something was coming from the ceiling, in that spot, passively accepting it. (00:22:10)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: While listening to Batman and Robin plan, the camera is close enough to the Penguin to see the prosthetic nose is differently colored. (00:07:00)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: When the Penguin orders his hook to be fired, it barely moves, then suddenly takes off. If you look closely at the hook before it moves, the connecting wire used to pull it out of frame can be seen. (00:13:30)
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)
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
Revealing mistake: When Batman throws the first punch at Burgess Meredith, he is hitting him with an uppercut so wide that it misses by a mile even the tip of his cigarette holder, it's incredibly fake even for the understandably low and cartoonish standards of this show. (00:21:30)
The Penguin's a Jinx (2) - S1-E4
Revealing mistake: Robin deals with the Penguin henchman in the elevator, then suddenly jumps on the counter, watches Batman conk heads (in a real fake way) and starts laughing like a slightly demented person. In all that, the guy in the elevator was still standing and looking like he was waiting for his cue. He is just suddenly KO with no explanation in the full view of the room a moment later. (00:21:50)
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