Ghost Bride/Gearold VS Jameo - S5-E25
Continuity mistake: In the scene where they are running from the second ghost bride, they go into a crypt and close the doors. One of the kids puts a shovel through the handles to keep it shut. In the next shot, it's gone. In the shot after that it's back.
Continuity mistake: Harold's age is inconsistent throughout the series. Harold has his Bar Mitzvah in "Harold's Bar Mitzvah", establishing that he is 13 years old, and in "Hey Harold!", he tells Patty that he was held back a couple years, which is why he is still in fourth grade. However, in "Helga on the Couch", he is shown in the same preschool class as the other kids, and in "The Journal", he is shown playing in the park with the other kids, and he is a baby just like all of them.
Suggested correction: It's possible that Harold wasn't the only student held back, even though he's older than his classmates. Just a thought.
So? How does that justify him being in the same preschool class as the other kids?
He was probably held back in preschool but was not mentioned.
Fair enough, but he is also in "The Journal" which took place before the kids were in preschool. He is shown playing with the other kids as a baby, and there is no sign of an age difference, even though he is supposed to be 4 years older than them at most. All the other main kid characters are around 9 years old in the fourth grade. It's highly unlikely that Harold would be around those kids as a baby.

Beaned / Old Iron Man - S5-E11
Continuity mistake: When Arnold reads to Helga, the book title says "GEO-" and "GRAPHY" (two words); the next day, it says "GEOGRAPHY" (one word).
Stuck in a Tree / Rhonda Goes Broke - S5-E10
Other mistake: In the scene after the garbage dumpster scene, Buckley's lips disappear briefly while he's talking.

Love and Cheese / Weighing Harold - S4-E7
Continuity mistake: Lila was pretty much entirely underwater in a "standing" position when Arnold rescued her, how come suddenly the water only reached her knees?

Continuity mistake: In "Miss Perfect", Helga, Rhonda, Sheena, Phoebe, and Nadine are cooking up ways to make Lila's (the new girl) life miserable. In one shot, Rhonda is wearing her new blue outfit. When the scene cuts to the boys and Lila, and then back to the five girls, Rhonda is wearing her standard red sweater and black pants. For the rest of that day, she is back in her new outfit.
The Aptitude Test / Oskar Gets a Job - S3-E9
Continuity mistake: In "Oscar Gets a Job", Oscar goes around asking people for money. When he talks to Ernie, you can see that there is a phone and a light to the right of the door. When Arnold confronts Oscar in the hall, you can see Ernie's room (7). The phone has disappeared.
Magic Show / 24 Hours to Live - S1-E18
Continuity mistake: The boarding house has a downstairs bathroom, but in "Gerald Comes Over", there's only one bathroom and it's upstairs.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: After all the kids run into Phoebe, when Sid runs past her and her hat comes off, we can see that the hair above her scrunchie is yellow instead of black as it should be. (00:03:44)

Abner Come Home / The Sewer King - S1-E16
Other mistake: Abner Come Home: When Arnold wakes up and starts looking for Abner, as he walks into the kitchen, his hair clips through the doorway. (00:04:14)

Abner Come Home / The Sewer King - S1-E16
Other mistake: Abner Come Home: In Abner's flashback, Grandpa shows him how to get in and out of the door. We see grass outside the door, but what we should actually see are the stairs that lead to the door, not grass. (00:03:08)

Continuity mistake: When Pea Pod Kid is taking Rhonda's marriage predictor test, Sid is walking down the stairs behind them. In the next shot, he disappears. (00:03:44)
Love and Cheese / Weighing Harold - S4-E7
Other mistake: Love and Cheese: When Arnold and Lila's boat is stuck at the edge of the waterfall, as Lila says, "Arnold, what will we do?" a piece of the boat next to Lila is incorrectly coloured like the background and the seat, so it looks like a hole in the boat. (00:10:14)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: When Helga is deciding on the new hall monitor and it shows a shot facing the crowd, many kids sitting in the back are faceless. (00:02:20)
Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: In the cafeteria, when Gerald throws his detention slips on the floor, Phoebe gives him another slip. Phoebe is first behind the railing, but in the next shot, she is in front of the railing. In the following shot, she is behind the railing again. (00:08:26)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: When Helga chooses Phoebe to be the new hall monitor, as Phoebe is walking to the stage, Iggy's hair is incorrectly coloured black instead of brown. (00:02:30)

Timberly Loves Arnold / Eugene, Eugene! - S5-E22
Other mistake: Eugene, Eugene!: The hat Lila wears with her Betty costume has a yellow crown with a green ribbon around it. However, when Arnold sings, "Nice Guys Finish Last," as he sings, "Take his money, grab his honey," the crown of her hat is entirely green. (00:17:00)
Timberly Loves Arnold / Eugene, Eugene! - S5-E22
Other mistake: Eugene, Eugene!: When the kids sing "Sunny Side Up" at the end, as they sing, "Who cares if the day is foggy and gray?", the crown of Lila's hat disappears for one frame. (00:22:46)
Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: The big beeper float constantly changes appearance during the parade. Some examples: The sign saying "Let's Go Downtown" on the front changes to "Let' Go Downtown" in one shot. The sign saying "Let's Go Downtown" on the back is missing in one shot. The orange base is sometimes blue. The blue city design around the base sometimes isn't there, and it even changes colour. (00:08:17)

Runaway Float / Partners - S2-E20
Other mistake: Runaway Float: When Bob asks the kids who wants to ride on the float, Arnold says to them it's great the way it is. In one shot, Iggy's hair is incorrectly coloured black instead of brown. (00:05:52)