When Riggs is trying to jump onto the bad guy's car he is half in & half out of Murtaugh's car but just as they round a bend, following the Grand Prix cars, he can be seen sitting on top of the car just before he jumps. A few frames later he has returned to the previous half in/half out position.
In the chase scene during the Grand Prix the forward shots from the point of view of the police car show dry tarmac and racing conditions, however backward shots show wet tarmac and the race cars dragging wet trails, clearly a sign that it rained during the shots.
Driving in the Grand Prix, shots looking back showed a wet track with spray but when the camera looked forward, the track was dry with tyre marks.
The red and blue lights are alternately flashing and not flashing during the chase of the Mustang.
As is often erroneously displayed on TV police shows, it is impossible to "lateral" transfer from an agency in one state to another. A detective would have to resign from the first agency and then be hired (likely as a patrol officer) in the new state.
A bus driver would not be charged with manslaughter for hitting a pedestrian that walked into the path of the bus.
Riggs is shown jumping off the Mustang and landing/stopping close to where he landed. The Mustang had enough speed to overturn after hitting the barrels. Due to kinetic energy Riggs would have tumbled for a long distance.
The final scenes are filled with various usual Hollywood firearm errors. In the warehouse, nearly every time the bad guys are seen, they are heard to cock their rifles, yet no rounds are ejected. People that use firearms carry them with a round in the chamber so they can be used right away. Similarly, the sniper is seen to cycle his semi-automatic rifle and then adjust the settings on the scope. Since it was a semi-automatic, it would have reloaded itself after he fired the first shot. Scopes are set up for a certain distance and then range is compensated by use of the cross hairs/reticle. No trained marksman would change settings on the fly as shown, and if he did, it would likely cause any follow up shot to miss.