Continuity mistake: When Professor Farnsworth places the envelope containing his holo-message on the table, we get a quick glimpse of the front of the envelope, and it is blank, without any writing on it. He then walks away with the Sunset Squad robot, without taking time to address the envelope. When the crew finds the message, the words "For my crew" appear on the envelope.
My Problem With Popplers (a.k.a. The Problem With Popplers) - S2-E18
Continuity mistake: When Lrr bends the iron bars on the cage Leela is in, the stage in front of him is empty. When he is about to swallow her, a microphone suddenly appear in front of him. (00:18:00)
Continuity mistake: When Bender is watching television with Fry, in the background you can see the dart in the background is on the top half of the board in the wall, but when the shot changes it is in the bottom half of the wall. (00:10:00)
My Problem With Popplers (a.k.a. The Problem With Popplers) - S2-E18
Continuity mistake: When they first land on the Poppler planet Fry walks to some bushes. When it cuts to a close-up of Fry the bushes' appearance change. (00:01:25)
Continuity mistake: As Fry is looking for the atomic tiara he's in Leela's room. Before he goes through the drawers there are three of them, but when it cuts there are two drawers. (00:14:50)
Put Your Head on My Shoulder - S2-E10
Continuity mistake: When Fry and Amy are kissing inside the closet the items on the shelf behind them change between the shots. This also happens later when they are breaking up in the closet. (00:07:10 - 00:13:30)
Continuity mistake: When the professor puts the empathy chip to Bender's head, his grip on the pincers change between the shots. (00:08:30)
Anthology of Interest I - S2-E20
Continuity mistake: After killing Bender, Leela puts the microwave sideways on the kitchen bench. When it cuts it's placed normally against the wall. (00:11:10)
Continuity mistake: When Bender makes the joke about where the professor would be without all the students who love and respect him, we see the professor sitting with nothing in front of him on the table. When it cuts, two glasses and a blue cup suddenly appear. (00:02:35)
How Hermes Requisitioned His Groove Back - S2-E14
Continuity mistake: Before Hermes does the limbo under the flying file holders he drops his white shirt off. When it cuts there is no sign of the shirt on the floor. (00:19:05)
Continuity mistake: In the Hall of Presidents, Bill Clinton's head is above Jesse Ventura and Jimmy Carter. When he starts to hit on Leela, he's now sitting above Gerald Ford. Later he shifts again, being above Carter and Ford.
Why Must I Be a Crustacean in Love - S2-E9
Continuity mistake: As Zoidberg looks around at the beach, he is suddenly up on his mound, and two females are suddenly in the scene without walking onto frame.
Xmas Story - S2-E8
Continuity mistake: While Fry is commenting on it being his first Christmas without his family, it can be seen that the door to the hallway is open. but when Hermes comes in, there's the sound of the door opening. Also, Hermes waves the two stacks of cards. The stack in his right hand has a red envelope in it that changes to white in the wide shot.
Xmas Story - S2-E8
Continuity mistake: When Bender starts serving the bird for everyone, you see a chair between Fry and Tinny Tim. In the closer shot, Bender is in between Fry and Tinny Tim where the chair should be.
Anthology of Interest I - S2-E20
Continuity mistake: When Fry asks the Professor his 'What if?' question there are two shots from behind showing Scruffy and one shot in front. From behind, Scruffy has a brown moustache and a grey hat, but from the front he has his usual grey moustache and brown hat. (00:14:20)
Continuity mistake: Zoidberg holds the airlock open with his claw while lying on the floor. As the door lifts up and opens Zoidberg is stood up and first through the door.
Other mistake: Bender wears a magnetic bow tie. However in series 1 episode 2 it's shown magnets screw up his circuits and make him sing songs. It's mentioned in a previous entries correction that it's only when the magnet is near his head. This isn't true as it's shown in others episodes that it happens when a magnet is attached anywhere on him.
Suggested correction: It's strong magnets near his head that mess up his inhibition unit. The bow tie is a very weak magnet.