Steely Phil / Quantity Time - S2-E12
Continuity mistake: The digits on Arnold's clock get bigger and change shape while he's talking to Grandpa.

Casa Paradiso / Gerald's Tonsils - S3-E3
Continuity mistake: Gerald's doctor's name is spelled "Murray Stieglitz" on the sign, but "Steiglitz" in the credits.

6th Grade Girls/The Baseball - S1-E5
Continuity mistake: While walking down the stairs, Maria only has three toes on each foot; she has four at the pool.

Benchwarmer / Cool Jerk - S1-E11
Continuity mistake: The scoreboard has room for 10 names on each side when shown in close-up, but when Arnold is shooting free throws, there are only 7 spaces on each side.

Downtown as Fruits / Eugene's Bike - S1-E1
Continuity mistake: In the Eugene's Bike episode, when Arnold has to tie his shoe, he sets his books down on a bike in a bike rack. This causes the bike to tip over and makes all the other bikes in the rack to fall down. Eugene's bike falls into the middle of the street and get ran over. When Eugene finds out his bike is destroyed, he looks at Arnold and yells. Right after this you get a good look at the bike rack. You'll notice there are no bikes in or around the bike rack.
Benchwarmer / Cool Jerk - S1-E11
Other mistake: The names for the Green and Yellow teams' players are the same.
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.

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)

Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: When Arnold reaches the parade float on the bike, Iggy's shirt is incorrectly coloured green instead of purple. It's also green when the float rides past Mayor Dixie. (00:09:45)

Runaway Float / Partners - S2-E20
Other mistake: Runaway Float: After Nick talks about redesigning the parade float, Bob asks the kids who wants to ride on the float. Arnold then tells them it's great the way it is, and when the kids start saying, "Me! Me!", there is an animation glitch where part of Robert's head and hands get stuck in the air for about a second. (00:06:01)