Corrected entry: In the sewer, the guy with the big forehead starts singing with his guitar and a crowd gathers around. But you can see Leela's parents standing in the crowd and having no reaction to their daughter whatsoever, despite them knowing what she looks like. Considering later episodes, this is a bit confusing.
Trivia: If you watch closely, after Fry is leaning on the chair just after the countdown to ten reaches zero and he loses his balance, they cut to a close-up of the chair's legs falling over. You can see Nibbler's shadow cast on the wall in front of him. This was however retroactively added to the first season DVD. In the original airing of the pilot on TV Nibbler's shadow was not present.
Suggested correction: This has proven to be in the original airing.
Roswell That Ends Well - S4-E1
Corrected entry: The sergeant refers to Fry's grandfather Enos by his first name, something that would never happen. A superior officer - especially a non-com - would refer to him as "Fry", "Private" or "Private Fry". No army sergeant ever referred to an enlisted man by his first name.
Correction: You've clearly never been in the military. First names and nicknames will get used, especially if they are unusual.
Correction: A Sergeant is not an officer, he is enlisted just like the privates. The one in question was probably an E6 or E7.
Saturday Morning Fun Pit - S7-E19
Revealing mistake: Near the end of the "Bendee-Boo" segment, while everyone is laughing, one of the Harlem Globetrotters' (between Amy and Leela) right arm keeps disappearing and reappearing several times in one shot.
Suggested correction: It's on purpose like the rest of the fourth wall Scooby Doo animation references.
Episode Two: The Series Has Landed - S1-E2
Corrected entry: When they launch to the moon, the ship takes off and arrives at the moon in about 1.5 seconds, at about twice the speed of light. As they are still in Earth's atmosphere, the ship would be, as a result, slamming into air molecules at twice the speed of light. Even at 90% the speed of light, something coming into contact with another molecule would fuse that molecule with the object and send out a huge burst of gamma radiation, tearing apart that molecule and filling the air with plasma. If such a takeoff was to happen in real life, the air molecules that the Planet Express Ship contacts would result in a plasma wall so huge that New York would be leveled, and the ship would just be dust.
Correction: As is specified in later episodes the Planet Express spaceship doesn't move anywhere - it bends space around it. Anyone trying to argue the physics of such a procedure is, quite frankly, wasting their time.
Correction: Given that there is centuries worth of technological advancement as well as tech from alien cultures, we cannot say that there aren't advances that make it safe.
Kif Gets Knocked Up a Notch - S5-E5
Continuity mistake: Zoidberg wasn't in the machine to begin with, yet he appeared later. When it spins you see the whole inside of the machine - he's definitely not there.
Suggested correction: They even make a comment about this, he lives there. So he was probably sleeping under the grate but still got ejected. There's parts of the floor we don't see.
Correction: This is not a mistake. At this point in the series, Leela's parents are still trying to keep their distance, and not let anyone know that Leela is a mutant.