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Continuity: 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: 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.