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.
In many episodes the Earth's knowledge of things that have happened in the past is lacking, due to an apparent loss of historical documents throughout time (for example, in Season 1 Episode 2 "The Series Has Landed" it is made clear humans don't know the exact details of the first moon landing, or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".)
Leela's biological father is a mutant; one of his traits is that when he speaks his mouth appears to run vertically and open horizontally, whereas it is the opposite for a "normal" person. The problem with this is that he goes back to having a normal, human-looking mouth when not speaking.