The slow-motion scene in which Riggs falls from the highway is filmed from two angles. From above, one can clearly see that he's falling towards a padded mat. From below, he falls through a pile of boards.
After Jack shoots Billy in the interrogation room, there is a big blood spot on the wall behind Billy. When Riggs, Murtaugh and Cole come in a minute later, the blood spot is gone. Additionally, Jack shoots Billy in the middle of his chest but when Billy slumps, there's no blood on the back of the chair.
In the very last scene of the movie when Riggs and Murtaugh are bantering back and forth about opening the car door for one another, the same yellow early 70s El Camino can be seen over Rigg's shoulder turning down the street twice.
When Riggs is demonstrating the cop-killer bullets on the firing range he shoots at a bullet proof vest several yards away. As the vest is being bought back to Riggs you can see the bullet hole where the bullet went through the front and back of the vest. This means it penetrated two layers of armor. At the construction site Lorna wears two vests. Travis shoots her from only a few feet away yet the bullet only penetrates one layer of armor, not two.
When Riggs is following the villains' pickup truck, they turn
into a highway and a plane is spotted landing, it is a Lockheed L-1011 Tristar from Delta Airlines. In the next shot, apparently on the same spot, the plane suddenly changes to an Airbus A300 from American Airlines.
Near the end of the armored car chase scene, Riggs causes one of the bad guys to fly through the armored car's front wind shield. These would normally be made of bullet resistant glass several times thicker than regular glass which would have stopped a human body without even cracking.
The motorbike Riggs has commandeered apparently knows about its own imminent demise. As it plummets to the ground, its siren changes pitch and dies out before it actually hits the ground.
At the hockey game when Rigs gets the idea to use the PA system, they are announcing a penalty on the Los Angeles Kings. Moments later when the Kings score, there are 5 LA skaters on the ice, but only 4 Toronto players meaning the penalty was on Toronto, not Los Angeles.
In an underground station under construction (or refurbishment), the running rails would not be live, so there would be no chance of electrocution.
The machine pistol used by Darryl and later Sergeant Murtaugh is not a MAC-10 as stated, but is in fact a Cobray M-11.
During the shootout at the burger stand, Sgt. Murtaugh's gun switches from a silver automatic to a black revolver. However, after Murtaugh empties his automatic he can be seen reaching for his secondary weapon, the black revolver. It's common for police officers, especially plain clothes officers, to carry a back-up weapon.
Roger incorrectly tells his son to shave "with the grain." This is because Roger is used to an old-fashioned so-called safety razor, which is not as forgiving as a modern razor and can dig in when shaving against the grain. Shaving "with the grain" or "against the grain" is ultimately a personal preference, even with modern razors, so Roger wouldn't be "incorrect" in his advice.
The security guard at the hockey rink is replaced by a dummy as he is roughed up.
The scene after Travis shoots Leo on the ice, and Riggs chases after him, when Riggs loses Travis in the maintenance corridor and starts kicking and cursing for losing him. Look closely, you can see the set painting drawn to look like the corridor continues much further out (look at the ceiling, you can clearly see the end of the set painting).
The motorcycle cop Riggs commandeers the bike from is already handing him the spare clips as he asks for them.
As Murtaugh turns from the motorcycle wreck/explosion, the last several frames of the shot show the cable supporting the motorcycle.
As the villains attempt to flee, they and Riggs drive onto a section of freeway that is being newly constructed. However there are already skid marks on the road surface, which is of course impossible. These appear to be from previous takes on the scene as they match the truck's path perfectly.
When Riggs jumps onto the front window of the Subway and flashes the driver his badge, the driver shouts out "Police!", but when you look at his reflection off the window, his lips don't move.
When Travis is telling Captain Murphy why he loves the police station in Murphy's car, Travis's reflection in the rear-view mirror shows that his lips are not moving.
Steadicam and operator visible when Riggs' motorcycle falls from the highway.
After Rigs grabs the parking ticket off his truck's window, the camera tracks over to Murtaugh in the passenger seat. As the trucks pulls away we can see the camera matte box with "Panavision" clearly visible as a reflection in the window.
During armored vehicle chase scene, right before Riggs unlocks door and enters cab, crew is seen strapped onto roof of armored car.
(At minute mark 25:23) The reflection of the car-mounted camera is visible in the scene in which Riggs and Murtaugh discuss Murtaugh's retirement.
When Riggs and Murtaugh start firing during the final shootout, camera equipment is reflected on the pile of wood they are hiding behind.
After Riggs writes on the form, and puts all of those medical procedures on the form, no one seems to recognize the fact that the handwriting is Riggs and not the doctors, this should be easily noticed by any trained medical staff, as they deal with the doctor everyday. This makes no sense and this should have canceled any medical procedures and the doctor could have possibly lost his job.
Riggs and Murtaugh are busted down to patrol after being blamed for the bomb going off, however, the bomb was on a timer and would have exploded anyway in the short amount of time it had left on the countdown.
After Riggs tackles the actor who was holding Rianne with a gun, it is so obvious something has gone wrong with the film, yet the director takes eights seconds before he yells cut.
After Lorna gets shot, Riggs is checking her pulse with his thumb on her throat. Anyone who's ever had first aid training knows this is a big mistake because the pulse you will feel is the one running through your own thumb. Pulse checking should always be done without the thumb. Riggs is a cop, therefore, he knows that.
In the bathtub scene near the end of the movie, Murtaugh says it's ten years until forced retirement and calculates that it's 3650 days, when in fact, it's actually 3652 days, as it includes two leap years.
During the final shootout, Jack Travis does not get involved until the guy wearing glasses runs to his office and tells him it's the cops they're burning everything. Over the previous few minutes, there were lots of gun shots and fires being lit, while Travis just stayed in his office doing nothing.
When Riggs shows his id to the various gang members and he makes a major mistake. Not only is this extremely stupid but Riggs as a veteran cop would know to call for backup, facing off against that many potential gang members would likely get him killed, and he just wouldn't do that. No cop would.