Revealing mistake: When the Batboat leaves the dock, heading out to sea, the blurred rotors of the camera helicopter are constantly seen across the top of the shot. (00:20:40)
Revealing mistake: Whenever the moving Batmobile is emitting flames from the afterburner, like when it leaves the Batcave for the first time, an object is hung below the exhaust port that resembles a camcorder. The gadget is part of the equipment added to make the car shoot flames while moving. During the TV series, flames are only seen in close-up of the car motionless, as this was a prop comprised of a copy of just the last few feet of the car. On TV, the car's rear license plate is visible. In this film, it's partially blocked by the added equipment. (00:03:35)
Continuity mistake: When the Batmobile arrives on the pier, it's parked with its front end facing open ocean (against the orange contraption on the dock) and its tail toward dry land. But when Batman and Robin return and phone the Navy, the car has reversed direction (tail now against the contraption). It's not likely Batman would arrive back at the pier, start the car, turn it around, park in the same spot, shut the car off, reopen the doors and then make a call, only to then drive away. (00:20:10 - 00:28:15)
Visible crew/equipment: As the Batcoper emerges from its hangar, multiple crew and equipment are reflected in the cockpit's glass bubble. (00:04:05)
Continuity mistake: As the Batmobile arrives at the pier, it turns into a parking space, one of two available between a green VW Beetle and a white station wagon. In the close-up as Batman and Robin exit the car, the spaces on both sides of the Batmobile are empty. Judging by the location of the "Moby Dick" sign at the edge of the pier, the white wagon has disappeared. (00:59:05)
Revealing mistake: When Batman brings the Penguin to the Batcave the Batmobile skids to a stop on the gray floor. It then begins to rotate in the next shot and it's now on a black platform. When the Penguin emerges from the car and walks toward the rear of it, the platform is obviously gone. (01:06:10)
Revealing mistake: All the rear shots of the Batmobile in motion show flames emitted from the afterburner. But in the shot of it leaving the pier, the flames go out a few moments before the shot ends. The effects equipment failed or ran out of fuel. (00:30:40)
Answer: The Riddler doesn't sing at all. He yells to the henchmen on board, "Prepare to repell boarders!" He does it in a almost musical way, true, but there's no song.
CCARNI ★