Well I'm not sure such a topic is really worth creating, being that The Simpsons is a fictional show and all, but I've always had a few things that I've been able to notice that don't quite make sense... as in, it leaves you thinking "That's not very realistic", or things that you just find hard to believe could really happen. Or it could just be simple animator goofs, as in one frame there's a picture on the wall of Grampa Simpson, and in the next frame it's gone or replaced with a houseplant.

One thing for me that I always found a bit hard to believe was the episode "Homer's Barbershop Quartet", obviously a good episode especially for me, since it's based on the rise and fall of The Beatles, but certain things just didn't add up. First of all, a question that I believe Lisa asked, was what happened to all the money that Homer, Barney, Apu and Skinner made? And why aren't they ever noticed anymore? If they were such a phenomenon, you'd think that they would have had more of an impact than just rising to popularity and losing it all in one summer. People would recognize them on the streets, and there personal lives would be covered extensively by paparazzi.

Any thoughts/ideas or more mistakes to speculate about are welcome

Another example: in the episode #4F24 ("Lisa The Simpson" - ), Lisa worries about her intelligence and discovers that every member of the Simpsons family gets dumb eventually (later she discovers that only boys are predisposed to that, i.e. Bart and Homer). But that contradicts completely with another episode, #BABF22 ("HOMR"), where we see that Homer is very smart, only the crayon in his brain makes him so dumb.

And another animation goof in the attachment. It's from the #AABF03 "Lisa Gets An A". Look at Lisa's smile...

Take a look at Marge's shoulder. I've never seen things like this before - looks like a wrong line to me, that was removed afterwards (but not completely).

ALM
LOL, the second one looks even scary to me.
But impressive examples indeed. "I really hope somebody got fired for that blunder!"

(this one is not by me, but from my Russian forum.) Remember Bart's birth. It wasn't planned, so Homer and Marge had to marry right after school when they discovered that Marge is pregnant. But right now Homer is 39 and Bart is 10, so when Bart was born, Homer was 29 (and Marge was 27)... a bit too old for finishing school!
In the "I Married Marge" episode, the one about Bart's birth, they both don't look like 27/29 either.

And another nice example I noticed by occasion today (see attachment, it's from AABF12 "Make Room For Lisa").
Milhouse's left foot overlaps the door

Wow... how could you notice that? O_o i had to look at the picture couple times and even then, without reading your description, i couldn't get it.
Do you play lot's of "Where's Waldo"s?

That moment is from Kiss Kiss Bang Bangalore. But how in the world did you find that one, Gary? It looks like that for only 1 frame. Same goes for you, Dag

We must have some diehard Where's Waldo finders here on the board.

Even simpler here - one member has posted this screenshot in the "Guess the episode" topic on my forum. I couldn't guess the episode, but the animation goof has caught my eye. Glad you liked it though.