(at around 1h 18 mins) As Percy is thrown into the air by the cyclops, he is holding the Fleece but while he is in the air he has nothing in his hands. It reappears the next shot when he is on the ground.
In the Washington DC scenes, there are skyscrapers in the background. The Heights of Buildings Act of 1910, limits the height of building, the tallest being 210 feet, not the blue glass building shown in the background.
Inside Charybdis Percy and friends find Clarisse and crew on a confederate war ship (the CSS Birmingham). The ship has a radar antennae rotating on top of the cabin. Confederate era ships did not have radars.
Polyphemus wears the Golden Fleece, which is said to have the power to heal or cure anything, on his shoulder throughout the film, yet his eyesight is still greatly affected by the injury he sustained from Odysseus, enough so that he mistakes Grover for a female cyclops.
After Kronos is dispatched and his followers flee, they leave his sarcophagus behind, which Percy and his friends neglect to destroy or take with them on their return to camp. The ending shows that it was simply left on the island, completely intact and unattended.
The Sea of Monsters is said to be outside of Poseidon's domain, which is why Percy cannot manipulate the water to help them escape from Charybdis. If this were the case, then his other water-based powers, such as his navigational intuition and healing abilities, should not have functioned properly, either.
When Percy and others are being swallowed down to the mouth of Charybdis, in one scene, all three of them are dangling off the side of the inflatable boat. In the next scene, they are all back in the boat and seated upright.
The camp is attacked by the Colchis bull, which makes a loud bang as it bursts out of a house - yet the giant mechanical beast was totally silent while it crashed into the rear wall, and then tunneled all the way through the entire building.
When the Colchis bull uses its flamethrower to attack Tyson, Tyson is engulfed with flames but comes out unscathed. He later explains to Clarisse that Cyclopes are fireproof. But that does not explain how his clothes and sunglasses were unharmed as well.