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.
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.