Revealing mistake: At the beginning of Rock Bottom, when the bus pulls up in front of Patrick and SpongeBob, right before Patrick steps up onto the bus, SpongeBob's right arm is missing and it suddenly appears as he starts moving.
Krab Borg / Rock-a-Bye Bivalve - S3-E8
Revealing mistake: In the shot that shows the outside of the Krusty Krab (in Krab Borg) they reused the drawing of right before the customers ran out of the building because you can see many customers crowded at the door. This shot lasts about 2 seconds but it's not hard to spot. (00:08:05)
Continuity mistake: When Krabs', Squidward's and Puff's ancestors attempt to flee with the coach they pass way more buildings, seen in the background, than there actually are as shown in the overhead view of the town.
Blackened Sponge/Mermaidman vs. SpongeBob - S5-E14
Continuity mistake: In 'Blackened Sponge' a flap door suddenly appears on Spongebob's bedroom door in the background as Gary goes away to laugh.
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.
Answer: I believe this is the Season 6 episode "Spongicus".