Continuity mistake: When Miss White is writing Mrs. Heimer on the chalk board, the camera shows Kevin/the class, when the camera comes back to her, she has a different sweater, hairstyle, and her name is written in a different spot. I'm not sure if this is some kind of flash to when everyone comes back to school after summer, or an error.
Continuity mistake: In episode 2.9, in the last scene we can see that Paul still does not play basketball well in the future, and future Kevin narrates that it is so. However, in episode 4.8, Paul actually becomes a better basketball player and makes the basketball team, which means that in the future he would not actually play in the same way or as badly as he did when he was younger.
Continuity mistake: In episode 37, "The St. Valentines Day Massacre", the writing on the card Kevin makes for Winnie changes from the time he makes it to when it's shown again at the end of the episode; he's shown printing it and writing just in one corner of the card, but when the card is shown again, the writing is in cursive and is spread down the side of the card.