These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Anvilicious - On at least one occasion, Lampshaded and mocked to death. Played straight many other times, though, especially in later seasons.
Some Anvils Need to Be Dropped: The finale, despite being depressing, did make a good point: We have to take care of the environment or we'll end up extinct like everyone on the show.
The episode "Baby Talk" makes a lot of good points about what can happen when Moral Guardians protest over every little obscene thing on TV and in real life.
"Out of the Frying Pan" delivers a nice Aesop about the perils of child stardom, as well as a nice Take That to stage moms everywhere.
Genius Bonus: Earl's job is pushing trees. Paper is made from trees, so he's really a paper pusher (a low-level office worker).
Harsher in Hindsight: In the episode "And the Winner Is..." Earl has a nightmare where the world ends as a result of him being elected as Chief Elder. In the last episode of the series, Earl causes the end of the world.
Somewhat averted in that the show actually began development before The Simpsons premiered. It was originally intended to be Jim Henson's follow-up to Labyrinth and would have had a more serious slant to it. Then The Land Before Time opened and Henson decided to go in a more comedic direction.
What Do You Mean, It's Not for Kids?: Dinosaurs was a lot like The Simpsons in its early days, in that the show looked like it could be for kids (The Simpsons had childish, crude animation and artnote and most of the episodes centered on Bart's and Lisa's problems in school and life while Dinosaurs had puppetry and animatronics), but the writing and subject matter both shows took on was very adult for something that looked like kiddie fare.
This was actually lampshaded in one episode. Earl is watching a sock puppet show on TV that noticeably has a lot of adult humor, but Fran dismisses it as just a kid's show. Earl explains to Fran that the sitcom may look like a kids' show because the characters are puppets, but it's actually for adults because of the witty dialogue that has a risque slant to it that older viewers will readily understand.