Episode Two: The Series Has Landed - S1-E2
Continuity mistake: When Fry and Leela are walking on the planet's surface, they have shadows. When they are next seen, the shadows have disappeared. The shadow inconsistencies happen again later when the buggy's wheel falls off and they try to outrun nightfall. (00:13:55)
Continuity mistake: At the end of the episode, when Bender and the Countess Delaroca are hanging on to the escape pod, the diamond on the diamond bracelet the Countess was hanging onto changes position between shots.
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: After destroying the toy spaceship Bender place the hammer on the ground in front of him. When it cuts the hammer has changed position. (00:00:55)
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: In the intro, before the main titles, the Professor informs the crew that he's booked them on the maiden voyage of the Titanic. When Leela says "Looks nice," at the beginning of the "scary" Titanic montage, the right side of her shirt is undamaged. During the middle of the "scary" Titanic montage, it suddenly becomes torn.
Continuity mistake: When Fry and Leela dig out Santa there are cracks in the ice around him which disappear when they lift him up.
I, Roommate - S1-E3
Continuity mistake: When the team are gathered round the table at the beginning of the episode, note their positions when the video screen is turned on. When they see Fry drying his hair, Leela and Bender have each moved up a chair, and Zoidberg has also moved, and seems to have taken half of his previous chair with him. (00:01:20)
The Honking - S3-E1
Continuity mistake: When the attorney reads up the will to the butler the attorney has no paper under his arm. When it cuts papers appear under his arm. (00:03:05)
Put Your Head on My Shoulder - S2-E10
Continuity mistake: When the tow truck appears to tow Amy's car back to Earth, the window is completely steamed. The tow truck driver only wipes away a small bit of the steam but in the next scene the window is completely clean.
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.
Plot hole: Whenever Leela arrives and everyone starts beating up Fry to make him cry out the emperor, it shows Dr. Zoidberg pinching his leg and later, sitting next to the wall, tired out with Bender and Amy. However, if he's there, then what is keeping the Trisolians from coming through the hole in the door that Zoidberg was plugging with his claw? (00:19:40)
Continuity mistake: In the clip before Fry knocks Fox off the air, the one person says, "Could you repeat the last part? *pause, cut.*" When we later see it broadcast on Omicron Persi 8, it says, "Could you repeat that last, *cut*."
Continuity mistake: When Leela tells Bender to stop destroying the earth she has her arms crossed. When it cuts they're down. (00:20:40)
I, Roommate - S1-E3
Continuity mistake: When Fry is watching "All My Circuits" for the first time, it shows Calculon screwing a golden nut onto the index finger of a fembot's right hand. In the very next shot, the bolt is on the middle finger of her hand. (00:03:15)
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.
Episode Two: The Series Has Landed - S1-E2
Continuity mistake: When Fry and Leela get on the moon buggy, there are three console operators and a gaffer standing up behind them. When they drive away, the gaffer disappears. (00:10:05)
Continuity mistake: The comic books pages become blank when Cubert is ripping off the back page.
Continuity mistake: When Bender switches to hard liquor, he takes out a martini in place of his current drink. This martini however is in the previous shot as he notices the setting suns. (00:14:25)