Factual error: Inspector Dreyfus is head of the Police Nationale and Clouseau is a gendarme (as stated several times in the film). The Gendarmerie is a military branch which conducts law enforcement mostly in rural France, while the Police Nationale is a civilian police agency responsible for urban areas. As head of a civilian agency, Dreyfus would have had no authority (or reason) to appoint a military member to any post.
Revealing mistake: When we see the gas going backwards into the canister during the Gas Mask Bandits attack at the Casino, you can see it is reversed footage.
Continuity mistake: When Clouseau is playing "good cop/bad cop", he offers cigarettes to the prisoner, which he declines. Clouseau puts them on the desk, where they promptly disappear from the remainder of the scene.
Revealing mistake: When Clouseau falls through the floor of Xania's room on to the hotel registration counter, you can see it is not Steve Martin but his stunt double.
Continuity mistake: When Clouseau is in LaRoque's office, you can see a table with his coat and a chair behind the desk before he hits the desk to break the vase off his hand. But when he hits the desk, the table and the chair have suddenly disappeared.
