Continuity mistake: When the bus passes Poseidon and he disappears, the people walking near him also disappear. There is an Asian couple walking from left to right across the shot that should have been still there after Poseidon is gone. Sloppy SFX that sort of spoils the effect of a god being able to vanish.
Continuity mistake: There are several scenes where Percy is shown wearing a watch, but it seems to disappear in the next shot or two (most noticable in the Parthenon scene).
Continuity mistake: Percy, Annabeth, and Grover escape in the sports car that was on display in the Lotus Club. It smashes through the casino doors and onto the street, but it emerges completely undamaged - no dents, scratches, or broken glass, whatsoever.
Continuity mistake: In the Medusa's scene, when the frightened woman got petrified, her right hand positions were different. First, the hand was in a horizontal position, but in close-up, the hand was in a more vertical position.
Continuity mistake: During most of the movie, Grover's legs have brown fur. But in Tartarus, Grover's legs has black fur and when Grover asks "What's different about me?" he has gray hair.
Suggested correction: In the movie (at least from the various times I have watched it) Grover never goes to Tartarus, please correct me if I am wrong but this just doesn't make sense.
Continuity mistake: At Auntie Em's Emporium her lip color changes between scenes from red to purple.
Continuity mistake: While Grover, Annabeth and Percy are at Medusa's. Percy has his sword out ready for combat. As he runs away from Medusa (with his sword in his hands) he bumps into Grover but Percy doesn't have his sword out.
Continuity mistake: Sally and Grover are taking Percy to Camp Halfblood; Sally has her head turned to the right, speaking to Percy. Grover, in the back seat, is looking front as he yells for her to watch out. Sally looks forward, and with a surprised look, swerves left to avoid whatever it is she sees. Then the cow, thrown by the minotaur, comes in from behind, above and to the right and lands in front of the car, a direction in which neither Sally or Grover were looking.
