Updates: Series 21
Continuity: In the bit where they sit on the sofa at the start of each episode, there is one rather long one where it starts off with the Simpsons dancing in a line, then gradually evolves into a circus with elephants and men dancing around fire. Look closely in between Homer's right leg and the dancing woman on his right. There is a pile of books on the little shelf underneath the lamp and telephone, and just as the camera starts zooming out from the line of dancing, the orange book on the left noticeably changes to purple.
Continuity: In the second segment of "Treehouse of Horror XXI," as Roger is telling them about what happened to him, as he finishes the story of Marge and Homer saving him, in his recount Marge is wearing a shirt and bikini bottoms (the shirt is covering the bikini bra). But when they saved her, she was just wearing a bikini.
Continuity: In the new opening sequence (season 20), you see Kearney and Jimbo sawing off the Jebediah Springfield statue's head, which starts falling down. In this wide plan, overhead and closing up, the whole town square is in full sight with no one visible besides those characters. The shot cuts to a close-up of the statue's head falling on Ralph Wiggum, now right next to the statue's pedestal where he was calmly eating an ice cream cone.
Continuity: When Homer walks out Circus of Values with his purchases, he bumps into Flanders, and all the presents scatter everywhere. Watch out for a red present with a green bow that goes flying near the left hand side of the screen. As soon as it hits the ground it turns into a pink box with a white bow wrapped around it, with no shot changes.