When the trainer comes around the corner after Clouseau hears the "heel" sound being made by the trainer's shoes there is a mismatch. The trainer's shoes were regular athletic sneakers, not cleats which could have made that noise.
When Clouseau is trying to park his car, he knocks off the front bumper of a cop car, and the rear bumper of a civilian car. In the shot of him coming up the steps, the cop car is completely different.
When Clouseau tries to download the first ring tone (right before the power cuts) he is clearly using the Mac OS X operating system with Safari. The next morning, when Clouseau gets set to download another ring tone, the computer's operating system is Windows XP with Internet Explorer.
When Dreyfus is hiding behind the drapes in Clouseau's office, both Dreyfus and Clouseau pass the computer on Clouseau's desk. When Clouseau slams Dreyfus's head on the desk, he slams it where the computer was.
The crucial "zoomed section" of the picture taken at the airport is clearly from a different picture than the one we are initially shown, as evidenced e.g. by the positioning of the characters in the background.
When Clouseau is working late and the secretary walks in and asks him if she should stay, the camera moves behind him, and you can see he is wearing a jacket. When the camera turns around, he is wearing only a white shirt.
Insp. Jacques Closeau and Chief Insp. Dreyfus mention several times that Bizu was shot in the occipital lobe. In the scene when Bizu was shot, he was facing the gunman, so he would have been shot in the forehead. The occipital lobe is in the back of the head, not in the forehead.
When the guard at the Presidential Ball confronts Clouseau and Ponton and asks them to prove that they are dancers, he cocks the hammer on his pistol.The pistol he is holding is a Glock which has an internal hammer, not external.
Since Clousau is in his apartment, its wiring must be part of the regular city electricity grid used for street lighting, so when the power goes off in his apartment, it should go out all over Paris, just as it does for the Eiffel Tower and all of the other lighted landmarks and major structures in the image, yet the regular city lights clearly stay on.
The goof item below may give away important plot points.
When Clouseau says that Xania may keep the Pink Panther, she puts it on her right hand, but in the next long shot, when he is explaining how he knew that she had it, she is still holding it, not wearing it.