Revealing mistake: When Fry starts his speech with "I, Fry", the podium is casting a shadow underneath him, but he isn't. (00:13:50)
Revealing mistake: When someone suggests crying as a way to get the emperor out of Fry's body, he says 'that's a great idea', the emperor's face stays still on Fry's stomach, instead of moving in sync with Fry's body. (00:15:50)
Revealing mistake: In the scene near the beginning where Leela tells Fry and Bender to watch the political debate on television, both of them (Fry and Bender) fall asleep. When Leela "hops" on the couch to wake them up, Bender's arm, despite being behind Leela, can be seen through her white shirt for a few frames.
A Bicyclops Built for Two - S2-E13
Revealing mistake: At the end of the wooing near the end, you can see that Fry has four arms.
The Honking - S3-E1
Revealing mistake: In the final battle, Bender has fry trapped into the seat inside of him, and he drives over a ramp and nearly smashes into the original were-car mid-air. But when the were-car lands on the same ramp that Bender took off from, we see that the furnace is literally touching the end of the ramp. How did Bender get up enough speed to launch himself into the air, especially accidentally (he would have had to turn 90 degrees to get onto the ramp) with the furnace that close? Also at the start of this scene, we see the were-car crying in front of the furnace, and the ramp is nowhere to be seen. (00:19:50)
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.