When Simone is holding George's graduation photo it has no border or white frame. Yet if you look very carefully when Charles hands it to her and after when he puts it under the ash tray on his night stand the photo has as a white frame border.
When Sir Charles is first skiing behind Princess Dala to spy on her movements, he is wearing binoculars with the strap over his right shoulder, his left arm through the strap, and binoculars under his left arm as he skis with ski poles in both hands. When he stops, his left arm and ski pole are no longer through the strap; the binocular strap is simply around his neck with the binoculars hanging on his chest as he grabs them to look through them.
The amount of champagne in the Princess's glass impossibly changes drastically between "just seconds apart" shots (i.e., where there would have been no time between the shots for her to drink more of the liquid or for Sir Charles to refill the glass) of her reclining on the tiger-pelt rug - in one shot it's nearly empty and she's tipping it back to drain the last of it from the bottom of the glass, and then in the next shot it is half-full, etc.
It is stated that the diamond has a flaw that resembles "a leaping panther", yet the image within the diamond is actually shown to just be the classic saucy "lounging back on his haunches and looking hilariously pert and adorably cute" pose of the panther that one sees at the beginning of every animated P.P. cartoon, not an image of him springing upwards or forwards.
At the start of his first visit with the Princess, Sir Charles Lytton is using his cane on the left. Within minutes, he's switched it to his right. This could just indicate that he's playing up the injury to spend time with her.
When Clouseau first opens the door to his room to look into the hall he yanks it open it swings halfway open then slams back into him. A small "stop" block can be seen fastened to the floor where the door can hit it. The block is gone in all other shots.
When the cab drives off with the Clouseau-impersonator who lost his slipper, the shadow of the rear of the departing cab recedes "upward" and out of the top of the frame, and then suddenly re-appears partway down into the frame instead of starting at the top of the frame and gradually working down into the frame again. Obvious "cut-and-splice" in the film between the footage of the cab leaving and that of it then hastily backing up again to let the villain retrieve his slipper.
The goof item below may give away important plot points.
When Clousea is led out of the courthouse in handcuffs, he is escorted by two Italian policemen through the crowd and into a waiting police car, which then drives off. However, in the next scene - an interior shot of the car with Clouseau in the back seat between the two officers - the actors playing the policemen are not the same ones who escorted him down the steps and into the car. One of them is a very recognizable William Bryant, a veteran American character actor.