Alternative Character Interpretation: Axel is a Manipulative Bastard and his various Bavarian Fire Drills on people with something he needs or wants are downright exploitative (he threatens to call a hotel racist in order to get a room without a reservation, then makes them give him a discounted rate. In the sequel, he steals a house undergoing renovation) and conducts illegal searches. Were it not for the much worse villains, he would easily be a Villain Protagonist.
Awesome Music: The film's soundtrack album was a bestseller, and for good reason, winning the 1986 Grammy Award for Best Score/Soundtrack in Visual Media. Standout tracks include:
"Axel F" by Harold Faltermeyer. An iconic theme tune, and catchy as hell.
Covered Up: Raise your hand if you first heard this movie's theme song from the Crazy Frog. It's okay, we won't judge you.
Designated Hero: We all love Axel. He is cool, cunning, will go the mile for his friends and would probably be a lot of fun to know. But that is us, the viewer, talking. In-universe this guy is guilty of numerous crimes. Breaking and entering, trespass, theft, assault, vandalism, identity fraud, and numerous dangerous traffic offences such as running a red light just to escape his police tail. Not to mention the fact that whilst many of his lies are reasonably harmless; in the first film he was perfectly willing to accuse an innocent hotel clerk of racism just to get a discounted room, and in the second film he literally stole a whole house. Put it this way: if you forget the first two sentences of this example and focus solely on his actions, he is a criminal who is helped immensely by having friends in high places and only forgivable because he is going up against even worse criminals than he is.
Inspector Todd. The guy playing him really was a cop in Detroit, so he knew the role he was going into perfectly and absolutely nails it. One of the third film's many problems was that they Dropped a Bridge on Him.
Serge, so much that they brought him back for the third.
Harsher in Hindsight: Gil Hill, the actor who played Inspector Todd, was a real-life cop who went on to have a career in Detroit politics. Hill was later accused of corruption and being involved in organised crime, drug dealing and attempted murder, and was under investigation by the FBI at the time of his death.
During his tirade at the Beverly Palms Hotel, Axel pretends to be writing an article called "Michael Jackson: Sitting on Top of the World" for Rolling Stone magazine. In real life, Playboy ran an article called "Eddie Murphy: Sitting on Top of the World."
In the climax between Axel Foley and Victor Maitland, Andrew Bogomil aids Foley in taking down Maitland after Jenny frees herself from Maitland's grasp. Three years later, Ronny Cox would suffer a similar fate to Maitland in RoboCop (1987).
Ho Yay: Mikey and Axel have some sizeable amounts of it going on. And Axel doesn't seem repelled at all by Serge's flirty nature.
Damon Wayans is the "Banana Man" whom Axel gets the crucial fruit from; notably he uses the Camp Gay voice/mannerisms he would later use as Blaine Edwards in the "Men on Film" skits on In Living Color!
Strawman Has a Point: Axel being a Cowboy Cop is depicted as being in the right compared to the By-the-Book Cop mentality of the Beverly Hills Police Department. Except almost everything in the movie shows how the Beverly Hills PD are right. Bogomil was absolutely correct to reprimand Taggart for physically assaulting Foley despite Foley brushing it off; if anything, that should have gotten Taggart immediately suspended at minimum. Foley's comments about how cops don't file charges against other cops have also not aged well.
Axel's Camp Gay schtick aged quite poorly after only 30 years of LGBTQ+ awareness. The voice and mannerisms are already rather groan-worthy, but the line about him having herpes is just downright cruel.
Taggart assaulting Foley is portrayed as a simple disagreement but was pretty bad even at the time and hasn't aged better at all, given that police brutality never stopped being a problem. Even if Bogomil's reprimanding of Taggart comes across as just establishing that the Beverly Hills cops do things strictly by the book, he was absolutely in the right to call Taggart out for this.
Axel's remark about how in Detroit, cops don't turn in other cops for law or rule violations, considering the huge backlash against police brutality and corruption in the 30 years since, which were made worse by the "thin blue line" of police officers refusing to testify against crimes committed by fellow officers.