Continuity mistake: When Hunsaker's shot, his eggnog gets all over his tie and shirt. When the camera angle cuts to Murtaugh's reaction, Hunsaker's shirt and tie are clean and there is no eggnog pouring from the container. Also there is no hole of any kind in the glass door behind him in which the bullet made. (01:04:00)
Suggested correction: You can see the eggnog on his shirt and tie. As Murtaugh is kneeling next to Hunsacker it is very visible.
Plot hole: One of the children is a witness that saw the man that planted the bomb in Dixie's house. He is said by a policeman to have been playing 'under the porch' and was close enough to make-out the Special Forces tatoo on the bomber's arm. Problem is, Dixie's house had no porch of any kind, and any other home's porch would've placed him too far to see such a small tatoo so confidently. (00:57:40)
Suggested correction: The rear of Dixie's house is never shown. The house could conceivably have a back porch the kid could have been playing under and the killer could have used the back door.
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.
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.