The Joker Goes to School (1) - S1-E15
Revealing mistake: Batman drives out of the batcave alone. The first bit of stock footage of the car leaving the cave does indeed show him alone in the batmobile. But the very next stock shot of the car speeding down the dirt road shows us Robin sitting in the passenger seat. (00:05:25)
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.
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)
A Horse of Another Color (2) - S3-E5
Revealing mistake: Ethel Merman just can't keep a straight face amidst Batman's campy silliness. Here and in the previous episode, she can be seen cracking up in several of her scenes with Burgess Meredith. (00:05:00)
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 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)
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)
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)
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)
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.
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)
The Riddler's False Notion (2) - S1-E32
Revealing mistake: The film can Riddler hands Van Jones supposedly contains his silent movie. But the hollow thump, obvious light weight and lack of the distinctive rattle a film reel would make, all reveal that the can is actually empty. (00:19:00)
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)
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)
Not Yet, He Ain't (2) - S1-E22
Revealing mistake: When being gunned down by O'Hara cops, there are no bullets coming from the cop who gunned Batman down, but he still reacts to it.
Revealing mistake: In the recap, in the wide shot where Mr Freeze uses his freeze gun on Batman and Robin, you can see they are stunt doubles.
Batman Stands Pat (2) - S1-E14
Revealing mistake: As Batman is about to use his batarang to Mad Hatter's hideout, right before getting caught by him and his henchmen towards the end of the episode, as he moves his cape back, the back part of his costume is open.
Revealing mistake: In this episode and the next one, Robin's stunt double is very obvious from the back, especially in the scene where they enter the Hall of Fame museum.
Batman Makes the Scenes (2) - S1-E34
Revealing mistake: While Batman and Robin are being held in Penguin's vacuum room, some of the stunt's dark squibs attached to the balloons become visible during this scene. We can even see a squib spark on the last pink balloon. (00:02:15)
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