True or False-Face (1) - S1-E17
Other mistake: When False Face lifts the wig off Blaze to reveal the crown that is on top of her head, you can see it is not there until the next shot of her.
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.
Not Yet, He Ain't (2) - S1-E22
Continuity mistake: When Batman and Robin are being chased by O'Hara and the cops, Batman raises his hand, goes to throw a smoke grenade, then in the next shot, he raises his hand again to actually throw it.
He Meets His Match, the Grisly Ghoul (2) - S1-E16
Other mistake: During the opening scene, right after Cesar Romero's guest star, Joker and Susie's faces freeze for a moment.
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 Stands Pat (2) - S1-E14
Other mistake: When Batman drop-kicks Mad Hatter into the vat of acid at the end of the fight, he does not have even one burn mark when he was pulled out.
Other mistake: During the final fight scene between the SOCK and POW, Mr. Freeze can be seen in the 76-degree heated section.
Batman Stands Pat (2) - S1-E14
Continuity mistake: Batman's plaster-covered costume gets an instant dry cleaning en route as he drives from the sculptor's studio to the batcave. (00:05:00)
Continuity mistake: When Batman and Robin are in the museum speaking with the director, Batman's chest emblem is high enough on his chest that it nearly touches the bottom edge of his cowl. In the next shot, when he leaves the museum, the emblem has moved down several inches.
Batman Stands Pat (2) - S1-E14
Continuity mistake: When Batman and Robin break through Mad Hatter's window, several large chunks of glass remain attached to the wood frame. In the very next shot, they've vanished. (00:18:00)
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)
When the Rat's Away the Mice Will Play (2) - S1-E12
Visible crew/equipment: The microphone dips into the shot at the end, just as Aunt Harriet is kissing Dick and Bruce good-bye. (00:23:20)
Smack in the Middle (2) - S1-E2
Continuity mistake: Molly is about to disguise herself as Robin. But the wig she's holding isn't even remotely the same color as Robin's hair. Of course, as soon as she puts it on and transforms herself, the wig darkens and becomes Robin's actual hair. (00:05:45)
Continuity mistake: While the unconscious Robin is being carried away from the Batmobile by one of Riddler's thugs, his cape loosens and appears to completely fall off just before they exit the frame. In the next shot, Robin is still being carried away but his cape is securely back in place.
Fine Feathered Finks (1) - S1-E3
Continuity mistake: When Batman reverses direction of the Batmobile via a "Bat-Turn", the car is shown from above turning counter-clockwise, but the next interior view shows the car is spinning clockwise. (00:14:20)
Audio problem: Joker's TV debut as an opera singer isn't very convincing. Even through the partially-obscuring clown mask, you can see that his lips aren't moving at all. (00:22:30)

Fine Feathered Finks (1) - S1-E3
Continuity mistake: When Batman starts the Bat-Turn, the parachutes are shown deploying on a 6-lane major roadway (3 lanes in each direction) in a very open area, but after completing the 180-degree turn, the car is shown driving away from the crumpled chutes on a much narrower side street among tall buildings. (00:14:10)
A Riddle a Day Keeps the Riddler Away (1) - S1-E11
Visible crew/equipment: During the first scene in Commissioner Gordon's office, the microphone shadow is outlined on the front of Chief O'Hara's uniform. (00:01:50)
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