Other mistake: Towards the end, when Riggs runs up to Murtaugh just after the car explodes, Riggs is carrying a large machine gun in his right hand. He runs up to the police barricade, but the police officer won't let him by until he shows him his ID. The police officer completely ignored the fact that Riggs was holding a large gun and could have been a criminal who wanted to use it.
Other mistake: When Ricky confronts the mother superior in the dining room, Ricky raises the axe above his head and brings it straight down in one single motion. Both Ricky and the mother superior are facing each other. At that angle, it would have been impossible for Ricky to decapitate her. At best her head would have been cleaved in half, but not decapitated.