Continuity mistake: In "Where's Gary?" the length of the paddle ball string that SpongeBob uses seems to vary in certain shots. While the length is somewhat subjective one shot that is more conclusive is when Patrick finds Spongebob lying on the floor. The string on the paddle ball is fairly short. When the two get up to get a glass of milk, the string is now a lot longer.
Continuity mistake: In "Bubblestand" after Squidward blows the huge bubble, he tells Spongebob and Patrick that the ability to blow bubbles is in his genes. If you look at the position of the bubblestand sign, the sign is facing the direction Squidward is talking. The sign had always been facing the street, which means that Squidward was actually talking towards the street, when Spongebob and Patrick are actually to his right.
Continuity mistake: On "One Krab's Trash", when Mr. Krabs draws a ghost on a sheet of paper, it shows before it goes into Spongebob's rooms that there's only writing on one side (i.e. the ghost) and when it goes into his room, a shopping list is on the back.
Continuity mistake: When Spongebob first shows the Patrick the technique for bubble-blowing, there are two red corals in the background - one on his right and one on his left. Later, when he shows Squidward the technique the red coral on the left side of the screen disappears.