The Penguin Declines (3) - S2-E39
Visible crew/equipment: When Venus is at the Batcave, loads of fisticuffs ensue between the Caped Crusaders, Joker, Penguin and all the baddies. While Robin is up on the atomic pile scaffolding he punches one goon off the ladder, then as he falls we can see the edge of the brown crash mat rise up, right before the word "SPLATT!" appears onscreen. (00:20:50)
The Penguin Declines (3) - S2-E39
Visible crew/equipment: When Batman and Robin bring Venus to the Batcave, just as she says, "It's fabulous," the reflection of the moving boom pole/mic is visible on one of the Bat-computer screens, at the bottom right corner of the screen. As this shot continues, when Robin points out their Bat-radar, a chalk T-mark is visible on the floor near Batman's foot. Then, when Penguin, Joker, and the goons appear, more chalk T-marks are visible on their side of the floor. (00:17:10)
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