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 Billy shouts at the Priest while holding the gun, there is a tripod standing behind Billy to his right. Billy yells, "Don't move." The shot changes to the Priest and then back to Billy again when Billy says, "I have a gun" you notice that the tripod behind him has disappeared. The tripod reappears in the overhead shot where the Priest says, "I'll explain everything."