Goof (continuity error): In Season 4 Episode 12 "Where No Fan Has Gone Before" Fry mentions the Star Trek films attracts negative attention because the show was banned long before then, and apparently saying the show's name is forbidden. In Season 3 Episode 8 "That's Lobstertainment!" at an Oscar awards ceremony a featured film is titled "Star Trek: The Pepsi Generation" and no mention of Star Trek being forbidden is made.
Goof (continuity error): In many episodes the Earth's knowledge of things that have happened in the past is lacking, due to an apparent loss of historical documents throughout time (for example, in Season 1 Episode 2 "The Series Has Landed" it is made clear humans don't know the exact details of the first moon landing, or in Season 5 Episode 2 "Jurassic Bark" it is unclear how a 20th century pizzeria operated,) yet many people from Fry's time period are still alive, such as Richard Nixon or Leonard Nimoy's head. Also, even though characters do not know the details of important events in or before the 20th century, many characters make references to popular songs of the past (William Shatner in Season 4 Episode 12 "Where No Fan Has Gone Before", Kif in Season 3 Episode 5 "Amazon Women In The Mood".)
Goof (continuity error): The elements Bender claims to be constructed out of change throughout the series (though given Bender's nature, it's possible he's lying). This became an in-joke amongst the writers in later episodes.
Goof (revealing mistake): It's dark outside in New York at midnight when Fry is frozen, but a long shot of the planet actually shows the eastern half of the United States as being in sunlight.
Goof (plot holes): The start of both the years 2000 and 3000 is heralded simultaneously across the globe. The midnights should be staggered due to time zones.
Goof (continuity error): When in the back room of the head museum, Bender pull out two bars to make an escape. But when he lifts them in triumph of his new options, in the window behind him the bars are back in place. On the next "shot", they're gone again.
Goof (factual errors): This episode portrays both 1999-12-31 and 2999-12-31 as "the eve of a new millennium". This is a common misconception; historical epochs traditionally have no year zero (the practice of starting an epoch at 0 instead of 1 came in in the 20th century, but the Anno Domini Epoch was established in 532 AD), so the correct dates for these landmarks are 2000-12-31 and 3000-12-31 respectively. (Those who claim that "the Third Millennium started on 2000-01-01" are in any case trying to have it both ways; they're counting years from 0 but centuries and millennia from 1.)
Goof (continuity error): Applied Cryogenics is part of "Old New York" and therefore should be buried under ground when Fry is unfrozen in year 3000 in "New New York".
Goof (errors in geography): When the world is counting down to the new year, we see people in various countries, Egypt, the USA etc all counting down together. Because of time zone differences this isn't possible.
Goof (plot holes): When Fry goes into the Cryogenic Freezer, the timer is set for 1000 years and goes in on January 1, 2000. Fry then comes out on December 31, 1999 instead of January 1, 3000.
Goof (plot holes): Fry mentions Star Trek, even though it is illegal in to say it in the future. So he should get arrested but doesn't.
Goof (plot holes): Since New New York was built on top of Old New York, the building the cryogenic lab was in should have been underground as well.
Goof (continuity error): When Leela calls for backup, her Wristlojackimator is on the wrong wrist. This happens when Fry and Bender are in the bar.
Goof (incorrectly regarded as a mistake): Leela and Fry find the site of an original moon landing but the lunar module is still intact. In reality the body of the module launched back into space leaving only the legs behind. However, it clearly states inside the lander "Lander returned to this site by the Historical Sticklers Society".
Goof (errors made by characters, possibly deliberate errors by the filmmakers): When outside the Apollo 11's lunar module, Fry compares his boot print to the famous print who he says belongs to Neil Armstrong. However, the famous boot print that he was comparing his boot to belonged to Buzz Aldrin.
Goof (continuity error): In the first snatch of "All my circuits", Calculon screws a "nutring" on Monique's forefinger. On the next frame, however, the ring is on her middle finger.
Goof (audio/visual unsynchronized): When the Planet Express Crew moves the couch out of the house, Hermes says, "We'll bill you for the couch" but his lips don't move.
Goof (continuity error): When Fry goes into the chemical shower the lights are in order with green on top and red on bottom; but when Bender comes back to Planet Express later on the positions have changed.
Goof (continuity error): In the judgment scene, when the judge freezes up, Fry yells: "Call technical support." In that frame, he is drawn with his manacle chain outside the bars, whereas in all other shots, it's on the inside with him.
Goof (plot holes): Fry has a box full of 20th century television shows on VHS. However, in the episode "When Aliens Attack", it is stated that all videotapes from that era were destroyed in the second coming of Jesus
Goof (continuity error): When Bender's cans first fall onto the floor, they don't have labels on them, but in the next shot they do.
Goof (continuity error): When Bender's cans first drop out of his jacket onto the floor, the number of extra items on the floor increases when the security cameras are looking down on him.
Goof (continuity error): When Fry has a bite of his pizza slice, the pizza slice changes size between the shots.
Goof (continuity error): Fry is seen opening the can of anchovies and putting one on a slice of pizza, but the rest of the anchovies are already on the pizza. This couldn't have been possible because that can of anchovies was the last can in existence.
Goof (factual errors): Bank accounts go dormant after 3 years of no activity and are turned over to the government as unclaimed funds, at which point no further interest would be earned. Activity can be something as little as checking the balance or simply receiving a phone call from the bank, but unless there has been someone doing this for the last thousand years, Fry's money would only have accrued three years of interest, not one thousand.
Goof (continuity error): As the suns set, Bender comments "At last, the suns are setting, finally I can switch to hard liquor." and gets rid of his wine glass in favor of a martini. But as he begins speaking, the camera angle is from a distance, and Bender is already holding a martini glass with an olive. When the camera zooms in, the has the wine glass that he throws away for the second martini glass.
Goof (plot holes): The law says Fry should be killed on the spot if he doesn't recite the oath from memory. He is reading it off his arm (and obviously so), that is not from memory; therefore, he should have been killed.
Goof (continuity error): At the inventions competition, when Ron Popeil introduces Ogden Wernstrom, Fry is sitting in the audience with his normal red jacket on. In the closeup, he is in a black suit.
Goof (continuity error): During the musical number, the Robot Devil pulls Bender's arm off and promptly kicks him down a hole. However when Bender lands his arm is still attached.
Goof (continuity error): Bender gets a symbol welded on to his body. When he welds something on to the spaceship, the symbol disappears, but reappears in the next scene.