Continuity mistake: In the hockey game, the teams playing are the Los Angeles Kings and the Toronto Maple Leafs. You can notice the real logos when the camera is away from the action. When it gets closer, though (probably because of copyrights), the maple leaf is turned into a spades (from cards) with only "Toronto" written and the Kings' word and crown in the logo become just an "LA" with a slightly different shape.
Continuity mistake: When Riggs is riding the motorcycle chasing the bad guys, the damage to the windscreen of the bike changes as the shot changes from close up to long shots and back. In the close-ups the windscreen is broken in half. In the long shots it is only cracked.
Continuity mistake: In the scene where the bad guys have taken the Captain hostage and are in the garage about to steal the guns, there's a closeup between the Captain & one of the bad guys talking. As the Captain turns, you can see an extra at the back wall dressed as a cop. He leans forward a little like he's supposed to jump in, then backs off. He's not there before or after that.
Continuity mistake: When the suspect is shot in the interrogation room, we can clearly see blood on the wall behind him. But when Riggs walks in and sees him, there's no blood visible.
Continuity mistake: When Riggs is following the villains' pickup truck, they make a left turn into a highway and a Delta plane is spotted landing. In the next shot, definitely showing the same turn (note the billboard in the background for "CBS FM" which is in the same place, etc.) the plane changes into an American Airlines plane - note the disappearance of the "Delta" logo and word at the front, and that the Delta plane has a single thin red line, whereas the AA plane has a blue, white, and red stripe along it.
Continuity mistake: The scene in the water after our two heroes have been fighting in the boat and Sheriff Dep. boat comes upon them: Notice Riggs's forehead hair - up, down, up, down... and both their hands keeps changing grip on the boat.
Continuity mistake: As the villains attempt to flee, they and Riggs drive onto a section of freeway that is being newly constructed. However there are already skid marks on the road surface, which is of course impossible, unless lots of people have screeched to a halt right at the edge of the freeway...
Continuity mistake: When Riggs and Murtaugh pull up at the construction site the windows on the car are clear glass, but when they send the car in without a driver the glass is suddenly black.
Continuity mistake: During the chase, an articulated lorry skids. If you look during one of the shots you can see that the trailer part is skidding over skid marks that already exist, obviously from previous takes.
Continuity mistake: In the underground scene, Riggs runs into the tunnel in trainers and come out wearing boots.
Visible crew/equipment: As the motorbike falls from the highway, just after it explodes there's a very short wide shot of the site with Danny Glover standing by his car and a couple of guys in hard hats standing around (or running for their life). If you look right behind Danny Glover you can see two people, one in a white hat, one in a red, and one of them is holding a camera - they start moving backwards just before the end of the shot. We then see the bike blowing up, then a close up shot of Danny Glover, from the camera we've just seen.
Other mistake: When Cole is looking at the LAPD records, Billy Phelps has a birth-date of 9/16/63 and is listed as 28 years old, meaning the current date is at least 9/16/91. However, the first record, Rick Sabman, has a birth-date of 9/1/60 and is only listed as 30 years old, when he should be 31.
Continuity mistake: When Riggs and Murtaugh are disarming the bomb in the car at the start of the movie, the sweat on Murtaugh's face keeps disappearing and re-appearing between shots.
Character mistake: The "MAC-10" machine pistol used by Darryl and Sergeant Murtough is actually a Cobray M11/9.
Visible crew/equipment: Right before the truck carrying ammo explodes near the end of the movie, it comes out of the house as Riggs dives for safety. A man can now be seen who should have been visible in the previous shot - looks like he is lighting a fuse and running... and then boom, the explosion goes off.