Factual error: In the desert shootout scene Murtagh throws what looks like an M67 "baseball" hand grenade on the ground. They wait for it to explode and Gary Busey says "it's a smoker". M8 smoke grenades look like a soda can, nothing like the baseball shape of a live M67.
Suggested correction: First, the grenade shown is more like the M61 "lemon" grenade. And there's no telling if Murtaugh/Riggs obtained a M61 based smoke grenade perhaps from a military surplus store that added the smoke effect for realism without the bang, or even a home made by Riggs grenade for this purpose, as opposed to using actual US Military M8 smoke grenades.
Factual error: When Riggs and Murtaugh are at the indoor shooting range, Riggs shoots a smiley face at long range. Every shot makes Mel Gibson blink and twitch from the bang. No expert marksman would react like that to the gunshot. Nor could his shot placement be so accurate with that reaction.
Continuity mistake: Towards the end before Riggs saves Roger and his daughter he is wet, but when he is running on the street his hair is absolutely dry.
Continuity mistake: When Murtaugh comes downstairs and checks his tie it's on the outside of his vest. He then walks through a door and the tie is on the inside of his vest.
Continuity mistake: After the car crashes into Murtaugh's living room, Riggs escorts Joshua out onto the lawn. As Murtaugh starts patting him down, a shot from behind shows Joshua with his hands behind his head. In the very next shot from the front, his hands are now lower, behind his neck.
Visible crew/equipment: At the tree lot when the drug dealer grabs Riggs, the shadow of the camera is visible on his chest.
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.
Revealing mistake: During the fight sequence between Mr. Joshua and Riggs at the end of the film, Mr. Joshua lifts up Riggs and carries him on his back before dumping on the hood of the police car. You can see that Mr. Joshua is not Gary Busey but a stunt double.
Deliberate mistake: At the end of the film, two officers attempt to surround and handcuff Joshua. Using two officers to cuff someone is NEVER done for the exact reason of what ends up happening, as Joshua manages to grab a gun from one of the officers. Only one officer would have been doing the deed in the real world, but it's clear this error in being completely against police protocol had to happen so the good guys could prevail over the villain.
Other mistake: When Murtaugh hangs the target at the shooting-range the hole is already in the center.
Continuity mistake: After Joshua is leaving Hunsaker's home after killing him, he is talking to Peter on a mobile phone and it is nearly dark. In Peter's scene in the car, the sun is still up and it is late afternoon/early evening. They are both in the same city.
Plot hole: There is no way Riggs and Murtaugh would have let Joshua kill the two officers guarding the house, if they were already waiting for him. And if they didn't get to the house first, there certainly would not have been enough time for them to reach the house and leave a message waiting for Joshua in that short space of time. So, were they waiting for him or not, and why let two cops be killed if they were?
Suggested correction: They didn't get to the house before Joshua; they showed up after he killed the two police officers. The note is in a child's handwriting, so it's more likely that Murtaugh called the family to tell them to evacuate the house, and one of his kids left the note.