Continuity mistake: When Fry and Bender are in the museum and locked in the "Villains" room, Bender breaks the two girders in the window, and then his arms falls off. As he puts his arms back in, you can see behind him that the girders are back in place in the window.
Continuity mistake: When Fry and Leela get to the moon landing site, Leela climbs in the moonlander hatch with the oxygen tank to which Fry's oxygen hose is connected. The hose is visible as it leads in to the hatch. When Leela pops her head out the hatch, the hose is suddenly gone. Then in the next shot, it's back, but lying underneath the hatch.
Other mistake: When the Moon Patrol officers throw Bender out of the airlock onto the moon's surface, they are exposed to hard vacuum, but they aren't wearing space suits. Throughout the episode it is established that the moon's "atmosphere" is a vacuum and protective suits are necessary - when the farmer removes his helmet he begins to asphyxiate instantly.
Continuity mistake: In the scene where Leela is telling Fry how even his food is recycled (just after they have seen the video of the origin of the garbage-ball) you can see both Fry's drink can and sandwich on the table. After the closeup of Fry his sandwich has disappeared but in the next full-table shot it's back (albeit moved to Fry's right along with his drink can).
Continuity mistake: At the Cryogenic facility, right where Fry shouts "Pizza delivery for..." the desk is positioned almost parallel to the window, but when Fry sits down at the desk, its positioned perpendicular to the window. And Fry can't have moved the desk because in both shots the freezing chamber is placed behind it so he would have had to move that as well.
Continuity mistake: When the spaceman is delivering the 'Planet Express' package, the colour of the rocket on the box changes from black to green. The same thing happens to the spaceman when he's picked up by the large bird - the logo on his backpack changes colour.
Plot hole: Whenever Leela arrives and everyone starts beating up Fry to make him cry out the emperor, it shows Dr. Zoidberg pinching his leg and later, sitting next to the wall, tired out with Bender and Amy. However, if he's there, then what is keeping the Trisolians from coming through the hole in the door that Zoidberg was plugging with his claw?
Continuity mistake: When Fry and Leela are walking on the planet's surface, they have shadows. When they are next seen, the shadows have disappeared. The shadow inconsistencies happen again later when the buggy's wheel falls off and they try to outrun nightfall.
Continuity mistake: When the team are gathered round the table at the beginning of the episode, note their positions when the video screen is turned on. When they see Fry drying his hair, Leela and Bender have each moved up a chair, and Zoidberg has also moved, and seems to have taken half of his previous chair with him.
Continuity mistake: After Zapp Brannigan shows off his 'big book of war', he returns it to a pouch at the side of his chair. Two shots later there is no sign of the pouch or the book. Also, when he gets out of the chair, there are no buttons on the armrests, but when he returns to it later, there are two on each side.
Other mistake: When Professor Farnsworth was presenting the crude drawing of the Smelloscope to his scientific peers, he wiped the napkin which it was drawn on on his forehead, causing all the ink to smudge. But why isn't there any ink on Professor Farnsworth's head?
Continuity mistake: When Bender sits beside Fry on the couch, Bender's sitting at the end with his arm on the rest. When Fry says 'Leela, we're trying to watch TV' Bender's moved to the middle of the couch. The garbage next to Fry and the contents on the table also change in this scene.