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 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: 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: 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: During the armored car chase, the bad guys' truck plate number ends "T2" while Delores' plate ends "E4". After Riggs tosses one bad guy out the back and decides to climb onto the outside, the trucks have switched sides with Delores curbside and the bad guy on her left. But two quick close-ups of the trucks bumping together shows Delores' truck is again on the left.
Continuity mistake: When Murtaugh and Riggs enter a parking garage to disarm the time bomb in a car's back seat, a No Smoking sign posted on the wall behind Murtaugh becomes obstructed by a vertical pipe and a red warning light between shots. At first the sign is plainly visible, but after Riggs slowly opens the door, the sign's reflection in a window near Murtaugh suddenly reveals the obstructions.
Continuity mistake: After the end credits roll there is a brief scene similar to the opening, with a building being blown up. Riggs and Murtaugh back out of the area just as the explosion starts and completely leave. There is a high angle shot and a police car appears where they just were. It wasn't there before.
Continuity mistake: During the chase involving the road-rail truck, the truck used by the villains is diesel powered, as evidenced by the "DIESEL" decal below the left headlight, seen while the truck is still on rails. However, later on when the truck is on surface roads, the below-headlight decal has changed to "EFI", indicating it is now a gas-powered truck.