The Simpsons is my favorite TV show of all-time - it's smart, heartfelt, and has an insanely talented staff of writers, voice actors, and animators that all worked together to create something wholly unique and special.
That being said, there are a few weird things that have nagged at me over the years. And not just weird animation things, like Smithers changing from black to caucasian or Ralph Wiggum's voice (and personality) abrupting switching, but more fundamental things in the plot and weird screw-ups.
It's not an uncommon trope to make the parents of a cartoon character look a lot like the kid - think Chuckie's dad on Rugrats, for instance. But to have BOTH parents look suspiciously like the kid is...questionable. Kirk and Luann Van Houten both look terrifyingly similar, to such a specific degree that there's almost no question they're related in the show's universe. From the bulbous noses to the specific red glasses to the blue hair, it'd be pretty surprising to learn they're NOT siblings who somehow accidentally engaged in incest.
Thankfully, it's not as bad as it could have been - originally voice actress Maggie Roswell, who played Luann, based her voice on that of Milhouse's. The producers thought it would be too weird to have Milhouse's mother both look AND speak like him, so they had her use a regular voice.
However, when Milhouse's father was later introduced, Hank Azaria tried doing a Milhouse-esque voice and the producers thought it was hilarious. So, it could have been worse, had the producers not weirdly rejected one voice actor's choice and then allowed another one to do the exact same thing.
UPDATE: apparently it was confirmed in a season 26 (!) episode that Kirk and Luann are indeed cousins. So...there's that.
In Bart's Comet (a seriously great episode), Bart is inadvertently admitted into Springfield Elementary's exclusive smart kid's club, the Superfriends, after discovering a comet by mistake. The Superfriends consist of the smartest dweebs in the school, from Database to Ham to Cosine to...Lisa. But there's one brilliant geek that's not there for some reason - the queen of summer himself, Martin Prince.
Why wouldn't Martin be included in the Superfriends, if only in the background? It might have been simply to preserve the joke of everyone having dorky nicknames except Lisa, although it wouldn't have been hard to claim his nickname was E-Mail or something. Martin's main joke is that he's the resident picked-on genius nerd that Bart only hangs out with out of sheer desperation already - so it seems a little weird he wasn't included.
The Simpsons go a long way to return to the status quo at the end of every week - the writers and producers actively fought against any real sense of continuity and often joked about how little consistency or character growth there was week-to-week, from Homer admitting he learned nothing by the end of Homer Badman to Homer going back to being annoyed by Flanders for absolutely no reason at the end of Homer Loves Flanders to...well, the friggin' Armin Tamzarian episode.
But one loose end that's never really referenced is the existence of Unkie Herb, Homer's half-brother - who got his vast fortune back AND repaired his relationship with Homer by the end of O Brother, Can You Spare Two Dimes? There's no longer any reason for Homer and Marge to not contact Unkie Herb when they run into money troubles (which they do - BIG money troubles, pretty often throughout the run of the show) - they're on good terms with him AND he owes his newfound fortune to their generosity.
For real - why would Homer ever go to Moe or Patty and Selma for his money troubles when he could just get some money from his super-rich half-brother?
Homer's primary joke characteristics are that he's dumb and grotesquely fat. Sure, there are other gags to be made about Homer - he's lazy, he smells bad (worse than melted hog fat, apparently), and he's a bad husband and an abusive father (which really isn't that funny, come to think of it). But again - the main ones are that he's dumb and fat.
The dumb one is hard to argue with - but the fat one comes into stark question every time Homer's canonical weight is brought up: 238 lbs.
Simply put, 238 lbs. isn't very fat, especially for Homer (who's supposedly 6' tall in the show). It's not slim 'n slender, but it's not the cartoonish level of obesity that defines Homer in the show. Hell, he apparently has an unprecedented level of body fat (based on Dr. Hibbert's jiggle-test).
Haha, boy - I really hope somebody got fired for that blunder.