When preparing to go to the moon, Fry asks if he can do the countdown, and Leela seems to barely know what he is talking about. However in the previous episode, while escaping from the police in the Professor's space ship, they do a traditional countdown from 10 (in synch with the upcoming new year). Though Leela may not be confused with the concept of counting backwards to an event, but because Leela is confused as to why Fry would want to have such an exciting build-up to a mundane activity like flying to the Moon.
When Amy is using the claw machine to get the keys back, she moves the stick to her left but the claw moves to her right.
The shadows on the ground should be much longer that close to the dark side of the moon.
Fry's oxygen tube disappears when Leela yells to him from inside the Lunar Lander.
Fry walks magically through his oxygen tube when he declares "It's every man for himself!" the second time.
Bender says he went to high school with a cardboard singer, but years later he says he is only 4 years old.
Bender's head is unscrewed counter-clockwise. But later, seen in the background, when he is securing it to his torso, the movement is also counter-clockwise.
The Redneck Farmer suddenly has a space suit on while chasing Bender.
Bender's beer bottle disappears from the table while Fry is talking about his breakfast.
When Bender says "A little" to Amy, the space around his eyes was painted Gray, in the next scene ,it changes back to black, it occured on other episodes.
While Prof. Farnsworth is trying to remove Bender's head, Leela's left boot is only a black silhouette.
After Zoidberg says "Excellent, excellent", he picks up something that looks like a little drill. He then asks Fry to open his mouth so he can look at his brain, and the drill disappears and does not appear in the rest of the scene.
When Fry and Prof. Farnsworth are outside Zoidberg's office, Fry appears way too tall.
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, but when Leela pops her head out the hatch, the hose is suddenly gone.
When Hermes is handing over the "Standard Legal Release" to Leela there is a red pen lying on the sheet. When the shot cuts, the pen is brown and is lying beside the sheet.
The United States flag vanishes from the landing site when Amy rescues the gang.
When Hermes walks to work in the morning, he jumps instantly from across the street to outside the front door.
Sound can't travel through empty space, which it happens to do so numerous times throughout the episode.
Amy's pocket where the keys go magically appears, then disappears twice.
Leela states that the flag is plastic, however it is actually a cloth flag (the only reason it doesn't collapse is that it is supported by a crossbar)
The Moon Landing site where Leela and Fry take refuge must be a reconstruction. For one thing, the flag should be on the ground; the Apollo 11 flag was knocked over by the rocket blast from the LEM when it lifted off from the Moon's surface (later missions placed the flag further away to prevent this). This leads to the more blatant inconsistency, which is that the upper part of the LEM really shouldn't be there at all; the astronauts would have used it to return to the command module. (not a goof per the episode, see photo)