In the opening scene when Riggs first tells Roger to take his clothes off, Roger is clearly seen with his hands on his jacket taking it off. Then it the next scene his hands are down and the jacket is back on his shoulders.
When Riggs is trapped underwater by a slab of concrete, Murtaugh strips off his coat and dives head first into the water. A trained policeman or emergency worker would always jump feet first, because of possible hazardous debris not visible from above.
When Riggs and Murtaugh enter the Murtaugh house they find the girls hostage. After the men drop their weapons, Riggs and Murtaugh tell the women to leave. They leave. After the brawl when they set the house on fire, the women are back in the house.
When Murtaugh and Riggs are running from the tanker explosion they pause to look before realizing the tanker is landing on their cruiser. But as they run away their cruiser that was completely shot up doesn't have a single bullet hole in it.
When Riggs and Lorna are driving together in a car right before the bad guys try to push them in front of the train you can clearly see that the car has a digital speedometer. When they are being pushed onto the train tracks the speedometer has changed to analog.
When Riggs, Murtaugh, and Butter are parked parallel to Leo, Butter is heard saying "License, Registration" softly in the background from an earlier cut before he redelivers the lines more loudly a moment later.
During the big car chase scene on the freeway, the table Riggs is on is missing the back right leg which is the same leg he eventually throws at the car, during subsequent scenes, the table sometimes has two legs and sometimes has three legs.
In the final shootout, Roger is holding a stainless steel semi-automatic pistol when he and Riggs shoot past each other at the bad guys. The action cuts away for a moment, and when it cuts back he's holding a six-shooter. When Roger shoots the guy that shoots Butters, he is using a blue-steel semi-automatic pistol. When Roger shoots one of the Four Fathers, he uses a revolver.
During the big car chase scene on the freeway, Riggs throws a table leg through the windshield of the bad guy, and the glass fractures around the hole. In subsequent shots, there is just a hole, and no fracturing of the windshield.
In the opening sequence when the tanker explodes, it begins falling back down to earth after flying just a couple feet off the ground, but in the next shot, it is falling straight down over Roger and Martin's car.
Captain Murphy arrives at the hospital and tells Riggs and Murtaugh they have to give up their captains' badges. He holds up two sergeants' badges, one in each hand. In his right hand is a badge on a black leather badge holder; in his left is a badge on a brown leather holder. In the immediate reverse shot, he is holding the brown badge holder in his right hand. Additionally, the badge is upside down (the open end of the clip is facing upward). In the following shot the badges are in their original positions.
In the final shootout, Roger tells Riggs he only has 2 bullets left. Then after Butters takes a bullet for him, Roger fires 2 bullets at the man who shot at him. Then he later kills one of the Four Fathers with his "final" bullet. He should have been out already.
In the railroad collision scene where Riggs and Lorna were coming to a halt, right before they do so, the car they were driving, a blue 1998 Ford Crown Victoria LX, is shown with a digital instrumentation cluster. In the climax were they were rear-ended, the next scene shows Riggs shifts the car into reverse. The car is shown to have an analog cluster.
During the chase on the freeway, Murtaugh drives his car very close to the back of the truck to help Riggs enter the trailer. Then he pulls his car to the left of the trailer. Just before he pulls the car, we can see there is a grey Oldsmobile in the left lane. However, when the camera changes and Murtaugh enters the lane, the Oldsmobile is not there; a new black car appeared.
When the Chinese truck is rammed by the second train, two cameras are clearly visible on the ground to the right of the train: black cloth-wrapped boxes with circular glass faces, mounted at 45-degree angles toward the vehicle, with stage lights behind them.
When Riggs drives up to his trailer home for the first time, the chrome plated front fender of his pick-up clearly reflects the feet of the steady-cam operator walking along the front of the car, plus a small tripod possibly used for a reflector.
After the scene where Lorna tells Riggs who the father of Murtaugh's grand baby is, when Riggs and Lorna pull up to Murtaugh's house and Riggs opens his door to get out of the truck, reflected in the door are several crewmen standing around in front of Murtaugh's house.
In the scene from the prison in China, as the Chinese men are being loaded onto a Boeing 727, the pilot inputs LAX (Los Angeles Int'l Airport) as the destination from Peking (Beijing), China. This would be impossible to fly on a Boeing 727 without stopping to refuel several times, as the 727 has a maximum range of roughly 3,000 miles - not enough to fly non-stop from Beijing to L.A., which is roughly 6,300 miles apart.
Also, in the scene showing the prisoners boarding the 727, we see a shot of the cockpit. In the center above the computer display, the engine gauges are visible in four columns. This would indicate that the aircraft has four engines, however, a 727 only has three engines.
The closeup shot of the printing plate was obviously created by using a dot screen - a printing process that creates variations in light and dark by varying the size of dots in a grid. Currency is created by physically etching in cross-hatched lines that vary in width to create these variation. The closeup shows a plate created by dots being etched as it if were an engraving.
After Riggs and Murtaugh have stopped "the human torch" and are walking off in the rain, Murtaugh is wearing only his boxers, due to the rain, it can be seen through his boxers that he's wearing black briefs underneath to avoid accidental nudity from being seen.
During the foot chase in China Town, the bad guy runs through the alley and after being blocked by Murtaugh on a bicycle at the end he turns to run up a fire escape. It is obviously a stunt man that is giving chase instead of Riggs. You can clearly see the stunt man look up the fire escape and chase him across the rooftops.
The goof item below may give away important plot points.
In the fight scene, between Riggs and Wah Sing Ku in the warehouse, near the end of the movie, when Roger "accidentally" shoots the Four Father you can hear a shell casing hit the floor. However, when it shows Roger, a moment later, he is clearly holding a revolver. Needless to say, a shell casing would only be ejected from a semi-automatic or automatic weapon.