Continuity mistake: When Fletch and Stanwyk first meet and Stanwyk asks Fletch to come to his home and hear his proposal, Stanwyk is wearing a suit/tie on top but is wearing jeans on the bottom. In the next wide shot as they leave the beach, and again in the next scene as they arrive at Stanwyk's house, he is wearing suit pants.
Continuity mistake: During the scene where Fletch is at the hospital trying to locate Stanwyk's medical records, he drops lots of files on the floor and asks a nurse to sort them out. When he is talking to her he has a stethoscope in his ears, but when the camera moves behind him they are suddenly round his neck. Moments later they are back in his ears.
Continuity mistake: During the scene where the cops grab Gummy under the boardwalk, the left passenger window of the cop car is rolled up as they open the door. After they put Gummy in the car, close the door and drive off, the left passenger window is still rolled up. Fletch grabs a rock and throws it at the cop car. Look closely at the shot where the rear window shatters - the left passenger window is now rolled down.
Continuity mistake: In the scene where Fletch is in the chief's office, he get thrown up against the wall of pictures. There are two pictures over his left shoulder and the top picture drops to the floor. After cutting to the chief and then back to Fletch, the picture is back on the wall making it two over Fletch's left shoulder.