When Alan and Dr. Kornbluth rescue Madison from the lab her tank is full of nasty looking cloudy water. However, when the scientists enter the lab to find Alan's brother sitting over the tank "fishing", the water is crystal clear.
After Allen and Kornbluth scuffle in the dentist's office and Kornbluth sits in the chair, the paper bib around his neck has been torn off. The bib is back in the next shot, then gone again as they leave the office.
When Kornbluth climbs onto the boat after the hose is pinched off, a regulator is attached to his tank. When the scene cuts to him after he berates the idiot twins, the regulator is missing from the tank. His wetsuit is still wet, indicating that he has just left the water.
When Allen's wallet drops to the ocean floor, you can see his "Charg-A-Card" with a "Vista" logo. When Madison takes the wallet to the shipwreck to look for a map of New York, the "Vista" logo is gone.
During the final car chase, Allen's BMW weaves around a taxi cab that has stopped in the intersection coming from the side. At the left of the screen you can see that the car also just misses a camera. This is the camera that captures the military truck going around the same cab in the next shot.
After Allen and Madison jump into the water at the end of the movie and fend off the frogmen, they immediately come upon a coral reef and other aquatic features that appear tropical and which would not be found in the Hudson River/New York Harbor.
Madison says she only has a week and must return when the moon is full. Twice when the moon is shown the phase is of about a week after full moon rather than a week before full moon. The dark sliver should be on the left rather than the right.
Allen hails a cab without his tuxedo jacket on, or in his hands, yet he has it by the time he arrives at Cape Cod. If you look closely, you can see the jacket tied around his waist; a buttonhole rests against his hip.
In the final car chase sequence, 2 men play chess in front of a clear walkway. When Allen and Madison pass them, a flock of pigeons suddenly appears, and rises into the air. When Allen drives in reverse to escape, the same pigeons fly backwards.
When Madison is in front of the televisions at Bloomingdale's, Allen asks what her name is. She replies in loud, screeching tones, shattering the televisions. When one television's screen breaks, its screen is clearly made of cardboard, not glass. It also shows the circuitry behind the picture tube, not the tube itself.
When the TV store is closing at Bloomingdale's, The Richard Simmons Show is shown on the monitors. In 1984, when the film is set, this show was broadcast only in the morning and was not released in syndication. The show would therefore not have been playing on a television channel in New York City in the late evening.