Revealing mistake: Right after they "create" Leeloo and she is lying in the plastic cage with the little straps on her, the guy approaches her and says, "if you want out, you are going to have to work on your communication skills." She gives him a dirty look, and then you can see a huge hole, pre-cut in the glass, that she busts her hand through.
Continuity mistake: When the cops go looking for Korben and Leeloo in the fog, there are two orange light sticks hanging from the ceiling inside their car. From the outside looking in they are swinging from side to side very quickly. But from the inside looking out they are swinging from side to side very slowly.
Other mistake: In the scene in the beginning when you first see the fifth element, and the camera moves up so you see the wide open mouth, look to the shoulder on the left. In the years of undisturbed dust, there are a few finger prints.
Audio problem: During any scene where weapons are being fired, the audio has the sound of cartridge casings falling onto a hard surface, but none of the weapons are ejecting cartridge casings to fall.
Continuity mistake: When Corben bails on the 5-0 (in his taxi with Leeloo), he yanks the police car's winch right out of the car (which resulted in dragging the winch, as well as the entire length of the cable behind his taxi). In the next shot showing his car (4 police cars chasing him down the "street"), the cable and winch have disappeared.