Continuity mistake: People (even the dogs) flying through the asteroid belt wear space helmets, but on the asteroids themselves, they don't.
Deliberate mistake: The fossilised dog looks like the dog did when he was young; it should look old and decrepit, as the dog was when he died. (Presumably deliberate, to avoid a spoiler.).
Continuity mistake: The lava is close to the machine, until Fry runs to it, then it's a long way away.
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.
Continuity mistake: The back page that has been ripped off is put in the envelope with the top part showing the right way up, but it was folded so if the ads side of the page was showing, either the middle part or the top part upside down would show.
Continuity mistake: The rotation of the feeding mechanism on the newspaper gun changes. It first spins counter-clockwise when Hermes starts firing, then it is shown spinning clockwise for the shot where Dwight takes over.
Continuity mistake: The comic books pages become blank when Cubert is ripping off the back page.
Continuity mistake: When Leela is found on the door step she has a bracelet on her wrist but in the next shot it is gone.
The Farnsworth Paradox - S5-E10
Continuity mistake: In the very beginning scene, where Fry and Leela are talking, there is an explosion. When some ceiling falls down there is one big bit and three small bits next to Leela's foot but then a few seconds later there is one big bit and a few small bits all in different places. (00:00:40)
Continuity mistake: When the crew and Bender are sitting on the couch, Bender says, " Oh, oh, oh! get ready for this part!" Note the vent behind him is missing the three lights that have always been on it. (00:13:25)
The Farnsworth Paradox - S5-E10
Continuity mistake: When Fry first takes the tangled Christmas lights out of the fake box, they are all red and blue. However, when the shot cuts outward to get a wider view, many of the lights have turned green.
Continuity mistake: Just before Leela unhooks her bra, she drops her singlet on the floor. The singlet she drops is a pale orange colour, matching her bra, but Leela's singlet is white.
Continuity mistake: The "Cartridge Robot" never takes the "Introduction" tape out of his slot, but in the wider shot it's suddenly vanished.
Continuity mistake: Bender is shown to be wearing clogs in the scene where he arrives back from Holland after saving a turtle. However, a few seconds later when he's knocked over, the clogs are gone.
Continuity mistake: When Bender and the other obsolete robots are watching a snail crawl across the beach, the snail is shown to be in the center of the screen with the robots standing around. Cut to Bender inserting a CD in his mouth, it then goes back to showing the original angle, but the snail has suddenly disappeared, and not enough time has passed for it to have crawled all the way out of the shot. (00:12:10)
Three Hundred Big Boys - S5-E11
Continuity mistake: Right before Leela dives into the whale tank, she drops her snorkel and mask, but when the shot pulls back, they've completely disappeared.
Kif Gets Knocked Up a Notch - S5-E5
Continuity mistake: When Zap Brannigan is about to shoot at the evil holograms the horse has fangs, but when they fly out the window the horse's fangs disappear.
Continuity mistake: When Bender is a "female" and is putting stuff in his locker, in one angle you can see a yellow item next to his pink handbag. Then in the wider angle, this item is gray.
Continuity mistake: When Amy and Leela are wrestling, Amy's leotard changes several times. It varies between pink knee-length leggings and no leggings at all.
The Farnsworth Paradox - S5-E10
Continuity mistake: Just as Hermes A is about to shoot the box into the sun, all the crew (A and B) shoot out into the airlock. However, Hermes B doesn't come out of the box and just appears in the pile. This is confirmed in the commentary.