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 Fat Jack's boat starts spinning around Allen, a cover appears over the motor that wasn't there before. In the next shot, it's missing again.
When Madison meets Allen on the beach, after rescuing him, her hair changes from wavy to straight several times before she re-enters the water.
When Allen and Dr. Kornbluth rescue Madison from the lab, her tank is full of nasty-looking, cloudy water. However, when the scientists enter the lab and find Alan's brother sitting over the tank "fishing", the water is crystal clear.
The moon is waning when it should be waxing.
Madison says she has only six days 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.
When Madison shows up at Liberty Island, she is completely nude. When the police take her inside the Statue of Liberty to move her away from the public you can notice that Madison is wearing clothing from the waste down.
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.
While diving, the scuba tank on Kornbluth's back is clearly fake. If it was a backup air source, the pinched-off air hose from the boat wouldn't have mattered.
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, it is clearly caused by a remote-controlled explosive, and the screen is clearly made of cardboard, not glass. It also shows the circuitry behind the picture tube, not the tube itself.
In the final car chase sequence, two 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 runs back into the water after kissing Allen on the beach, she has tan lines on her back, even though she had never been out of the water before, suggesting that Daryl Hannah had been wearing a bra before shooting that scene.
When Madison is exposed as a mermaid and dragged into a car, her legs and knees are visible as shapes in the fishtail.
When Madison enters Bloomingdale's, Ron Howard is reflected in one of the mirrors.
In the final sequence, bubbles from safety diver are visible ahead of Madison and Allen.
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.
During the final sequence in which Madison rescues Allen from the divers, she uses her tail twice to defend herself. In both instances, a hand is seen positioning a tail prop against the divers.
Before Allen and Madison leave for the dinner with the President, Allen presses the up button on the outside of the elevator. They are on the fourth floor, so Allen should have pressed the down button.