Picture Day/Pat No Pay/Blackjack - S5-E13
Continuity mistake: In 'Picture Day' the second time Spongebob paints on the stripes on his socks they disappear in the shot where he paint rolls on his shirt/pants.
The Algae's Always Greener / Sponge Guard On Duty - S3-E7
Continuity mistake: The Algae's Always Greener: At the start, the old man is carrying a tray with just a Krabby Patty and fries, but after the old man gets the ketchup bottle Plankton zooms in on the tray, and in this closeup there's a fork and a drink with a straw beside the Krabby Patty, then they vanish.
The Algae's Always Greener / Sponge Guard On Duty - S3-E7
Continuity mistake: The Algae's Always Greener: After Plankton uses his machine and wakes up in the Krusty Krab office, he looks out through the porthole on the office door leading to the eating area, but the porthole on that office door vanishes when Plankton returns to the office with the two Krabby Patties. (00:04:25)
Banned in Bikini Bottom/Stanley S. SquarePants - S5-E20
Continuity mistake: Banned in Bikini Bottom: After the police raid the Secret Krusty Krab, Krabs and SpongeBob are both in handcuffs, but when Miss Gristlepuss trips and the Krabby Patty falls into her mouth, both pairs of handcuffs have vanished, then the handcuffs reappear again on Krabs. (00:09:45)
Atlantis SquarePantis - S5-E12
Revealing mistake: Atlantis SquarePantis: After the Atlantis van crashes through the museum's ceiling, Squidward is surprised that it's the magical path to Atlantis, and while Krabs says, "Nice hot rod flames," note how only the disembodied tip of Patrick's arm is in front of SpongeBob's body (it even moves as Patrick moves), due to an animation error in this shot. (00:09:55)
Banned in Bikini Bottom/Stanley S. SquarePants - S5-E20
Continuity mistake: Stanley S. SquarePants: When Stanley's first customer tries to place an order Stanley pulls out a camera on a tripod, and in the next shot as Stanley hugs the customer for their first picture, the kitchen door behind them does not have the usual porthole, instead it has an anchor on it just like the door to Krabs' office. (00:19:15)
Hall Monitor / Jellyfish Jam - S1-E7
Continuity mistake: Hall Monitor: After SpongeBob walks out of class the bell rings and he gets trampled, and when Mrs. Puff follows him outside the classroom door is wide open, but then the entire door suddenly vanishes.
Naughty Nautical Neighbors / Boating School - S1-E4
Continuity mistake: Boating School: When SpongeBob remembers that it's the day of the boating exam, the calendar reads "boating exam today," but then it reads "driving exam today."
Banned in Bikini Bottom/Stanley S. SquarePants - S5-E20
Continuity mistake: Stanley S. SquarePants: When SpongeBob introduces Squidward to his cousin, Squidward's moai house has a doorknob at the right side of the door, but when Squidward loads the back of the moving truck the doorknob is at the left side of the door. (00:14:15)
Banned in Bikini Bottom/Stanley S. SquarePants - S5-E20
Continuity mistake: Banned in Bikini Bottom: After the chief of police tells his wife that he's starving he speeds off, and when it cuts to a closeup of the lock and chains, the wood plank directly behind the chains is crossed from upper right to lower left, but in the next shot it's crossed the opposite way. (00:05:40)
Patrick SmartPants/SquidBob TentaclePants - S4-E8
Plot hole: In 'Patrick SmartPants' Patrick lost the top of his head after falling off a cliff, but Spongebob put it back on. Suddenly Patrick's intelligence skyrocketed, but wanted to return to his former self in the end of the episode. To do this he re-enacted his fall, but it was unsuccessful. They eventually figured out that Spongebob had in fact not put on Patrick's head, but a 'brain collar'. How did Patrick posses his memories throughout the 'intelligent phase' despite not having his own head (and brain)?
The Snowball Effect / One Krabs Trash - S3-E3
Other mistake: In 'The Snowball Effect' when Squidward attempts to make Patrick and Spongebob resume their snowball fight he throws a snowball at them which they avoid, and it puts out Squidward's fire inside. Squidward had his back towards his house and threw the snowball in the opposite direction.
You Wish ('Shanghaied') / Gary Takes A Bath - S2-E11
Deliberate mistake: Spongebob gets stuck in the top of a tree as Gary removes the ladder. In previous episodes it's quite clear that he's more than willing to 'fall' as he takes no damage when hitting the ground due to the low fall speed.
Other mistake: At the very end, Sandy puts Wormy inside of a glass jar. Not only does she break the bubble surrounding it, there's no way for water not to enter the jar as she holds it sideways. Wormy should be killed.
Welcome to the Chum Bucket / Frankendoodle - S3-E2
Continuity mistake: The pencil's length in 'Frankendoodle' varies in between above and beneath the water surface.
Deliberate mistake: Spongebob blew a bubble to capture the butterfly while wearing that fishbowl on his head.
House Fancy/Krabby Road - S6-E1
Continuity mistake: Plankton's size changes drastically throughout Krabby Road. Plankton can be seen as reaching up to Spongebob's waist at one point. When he's clutching to the neck of a guitar, the amount of frets that go into his height change between one, two, three, and five, depending on the shot. (00:02:10)
Continuity mistake: When Mr. Krabs and SpongeBob are standing in front of the bomb after it lands but doesn't go off, the clock inside Mr. Krabs' office can be seen from the kitchen. The time on the clock is 5:00, but in the next shot, the time on the clock is suddenly 3:00. The bomb is then revealed to be a mechanical hand, and the time on the clock is now 5:00 again.
Continuity mistake: When SpongeBob accidentally reveals the location of the formula, Plankton drops the feather he is holding and lands in front of him. The feather remains on the floor, but when Pearl walks into the kitchen, the feather has suddenly vanished, and reappears in front of Plankton in the next shot.
Continuity mistake: When Plankton is telling Karen, "There's a whale in the laboratory," he is jumping up and down with his arms in the air, but in the next shot, his arms are suddenly down at his side.
Answer: I believe this is the Season 6 episode "Spongicus".