Factual error: The street signs in Beverly Hills are green on traffic signals, not blue as shown in the film. Blue signs are in Los Angeles, where the movie was filmed.
Factual error: When Axel first drives from the airport (while on the phone) towards Beverly Hills, he is coming from the wrong direction. He is shown on the interchange from the 105 Freeway westbound to the northbound 110 Freeway heading towards Downtown LA. If coming from the airport to that same intersection, he would be on the 105 eastbound into the 110 - not westbound.
Continuity mistake: When Axel introduces Serge, he places his hand next to his ear as if he were hard of hearing. From the opposite angle, his hand is lowered. As of then, the position of his hands is not continuous between shots.
Character mistake: When Axel is being arrested by the female officers, he says he's been a cop for "thirty years, and black for longer than that." Axel has been a cop for at least 40 years; the first film took place in 1984.
Other mistake: When Abbott is talking to Axel for the first time, he is flipping through a book that has files and pictures on Axel. One of them is a picture from when he got arrested after he got thrown out of the window. It is made to look like it was a picture taken of him at the time of his arrest, but it is a direct screen grab from Beverly Hills Cop.