Revealing mistake: Riggs handcuffs himself to the depressed businessman. They jump off the building but they're not handcuffed to each other anymore. They used plastic cuffs which could separate during the stunt in case anything went wrong. They break just as they jump, so they grab each other's hands to stay in contact. After they land the cuffs are attached again.
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.
Continuity mistake: The Christmas illumination on the balcony of Amanda Hunsaker's hotel room is arranged in three different ways: In the first shot, before she takes drugs, it is attached to the side of the railing. When she steps out, the lights are hanging about 10 inches below the railing, like they are later when Murtaugh searches her room. And when she climbs up to jump the bulbs are directly hanging down from the railing.
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.
Continuity mistake: As Riggs exits the car, he takes the cigarette out of his mouth. The camera angle chages to show him standing outside the car, handling his pistol, and the cigarette is back in his mouth, with no time to put it there and the audio indicates the shots are consecutive time.
Continuity mistake: While working out the case at the shooting range, Murtaugh only fires one shot, and Riggs fires 9 (offscreen shots during their continuous conversation are noticeably quieter, from other shooters). When they reel in the targets, Murtaugh's target has 6 holes, and Riggs' has 14 or 15.
Continuity mistake: When coked-up Amanda first looks down from the balcony, we see the ground from her vantage point. Were she to jump from this location, she would land on the huge awning below (the curved dark area). If she put real effort into jumping forward to clear the awning (and she didn't), she'd land just to the right of the lighted sidewalk, onto the grass. And if she were an olympic long-jumper and not high on cocaine, she might have landed on the red or black car parked at the curb. Under no circumstances could she have landed on the gray car, well to her left - yet she does.