Much like The Simpsons, this show (despite being a cartoon for older audiences) has dodged several censorial radar bullets back when it was on FOX (not so much the made-for-DVD films and the Comedy Central episodes, as the made-for-DVD films run on self-censorship [meaning the show writers and creators get to choose what's acceptable and what isn't] and Comedy Central is more liberal in what's allowed on their shows than FOX is).
From "Fear of a Bot Planet":
Bender (standing up): Hey, buddies, who wants popcorn?
(noises of approval from Planet Express crew)
(Bender stands for a few seconds as popping sounds reach to a crescendo and he inexplicably holds a popcorn container to catch falling popcorn from his compartment)
Bender (through mid-torso shot): Anyone want butter with that? (Bender then proceeds to press his antennae three times, which makes a squirting sound)
From "Why Must I Be a Crustacean in Love?"
Fry: Co-ed steam rooms. I LOVE the future.
Leela: Uh, Fry, you're in the women's steam room.
Fry: Ahhh. Fu-tur-istic.
Amy Wong: Psst. Look what life was like before genetic engineering.
Leela: Those poor 20th century women.
When the show was on FOX, the writers fought hard to have that scene kept in, as the censors at the time wanted it cut because of the nudity. (This preceded the nudity in all future episodes, with the possible exception of the made-for-DVD movies, as the crew relied more on self-censorship rather than Executive Meddling).
Fry: Stem cells? Aren't they controversial? Farnsworth: In your time, yes, but nowadays... shut up! Besides, these are adult stem cells, harvested from perfectly healthy adults, whom I killed for their stem cells.
Even more Refuge in Audacity comes from Farnsworth slapping Hermes' bottom to stop his crying, the way infants were traditionally slapped. When Amy comes out, she's fully conscious but the professor slaps her anyway from just off-screen.
Amy: *admiring her skin* Ooh, baby soft. Farnsworth: Indeed.
From "Proposition Infinity":
Bender: We're going to legalize it right here!
Hermes: Yeah! You've got to legalize it!
Amy: We're talking about robo-sexual marriage.
Hermes: We're talking about lots of stuff.
From "When Aliens Attack", when preparing to film an episode of "Single Female Lawyer"
Fry: Camera One!
* Cut to Bender; one eye pushes outwards, makes mechanical whirring noise*
Fry: Camera Two!
* Cut to Bender; his other eye pushes outwards, makes mechanical whirring noise*
Fry: Yes! I never thought I'd escape with my doodle, but I pulled it out.
Bender(offscreen): Just like at the movie theater! WOOOOO!
From "Bender's Big Score":
Hermes: That's why they call me eleven-inch Conrad.
After Dr. Zoidberg sews Farnsworth's hand back on, Hermes comments:
Hermes: You do a nice hand job, Zoidberg.
"Fun on a Bun" gives us this, as Fry and Leela are kissing:
Leela: Your head is returning to normal now.
Fry: The blood must've rushed somewhere else!
In "Mars University," Fry mentions attending Coney Island Community College and says their mascot was the whitefish. A "Coney Island Whitefish" isn't a fish at all; it's a slang term for a used condom found on the beach.
In "War Is the H Word," after defeating the ball aliens, there is a newspaper headline that says, "Balls Thoroughly Licked." The headline the writers originally wanted to use was "Earth Licks Balls," so the Executive Meddlingactually resulted in a much dirtier line.
Bendin' in the Wind: When Fry, Leela, Amy, Zoidberg, and Professor Farnsworth are in the old VW Beetle Bus, Farnsworth asks about a device that can speed up and slow down time. Fry answers by holding up a bong. The reason it got past FOX's radar? The script didn't refer to it as a bong, but as a "weird-looking bottle."
Even the comic books are not immune. In the story "Freaky Fry-day", a woman biologist introduces herself to Fry as Dr. Labianca.
Not necessarily an anatomical reference. It is an actual name, two of the victims of Charles Manson were Leno and Rosemary La Bianca.