Plot hole: Before Riggs, Murtaugh, and Rianne enter the disco after they have killed or chased away their torturers, a man is casually leaning at the bar, with his hands folded in front and moving with the music. When the three protagonists come running through the door he immediately turns around and points a gun at them which apparently he was holding in his hand. One should expect that, in a public disco, even the bad guys keep their guns somewhat better hidden than just in their folded hands when they are not aware of any immediate danger. Furthermore, even the guests close to the bar don't hear the shot when Riggs shoots that guy, even though the music is not loud enough to cover the gunshot. (01:25:20)
Revealing mistake: In one of the first scenes of the movie you see Riggs in his trailer. He walks out of bed smoking a cigarette and taking a beer. When he takes his first drink of the beer some fluid is spilled. If you look at it, it is not beer. It's probably water, but definitely not beer. (00:07:00)
Other mistake: Just after Danny Glover's shot the guy by the pool, Mel Gibson goes to handle the two girls. We hear all the exchanges between him and them offscreen, such as "don't move or I'll break your arms", etc., ending with him saying "Got 'em", and we hear a pair of handcuffs click. When we see the girls, they're either side of a tree, with the camera-side cuffs on a wrist each. But we only heard one set of cuffs (all while off-screen, so there's no reason we wouldn't have heard another set go on), and with only one set on, there's nothing to stop them running away. (00:40:20)
Factual error: After the car explosion in front of the Vogue theater on Hollywood Blvd, Murtaugh tells Riggs that "he's headed for the freeway" and "you can cut him off at the 3rd St bridge" (approximate wordage), and Riggs takes off running. 3rd St is nowhere near Hollywood Blvd (17 blocks away) and there is no 3rd St bridge within miles.
Continuity mistake: When Riggs is counting out of the money to pay the three drug dealers, he puts a 20 dollar note on the table. The wind then blows the note against the bag of cocaine and it is lying flat down, but in the next shot the note is folded up.
Audio problem: Right after Riggs handcuffs the suicidal man, the next shot shows Murtaugh saying "oh no" but his lips don't move.
Continuity mistake: When Hunsaker is talking to Murtaugh in his house, the shadows outside dramatically change throughout the scene. First the sun is beating down on Riggs from the right (over the ocean), then from the left leaving Riggs under a bunch of shadows from the tree. This constantly changes throughout the scene.