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.