
Mud Bowl / Gerald Moves Out - S2-E9
Continuity mistake: When Gerald catches his little sister destroying his cassette tape, the book she is wrapping changes color from blue to green. (00:12:30)

The Little Pink Book/Field Trip - S1-E2
Continuity mistake: On the "Presidents of the United States" sign behind Helga, the word "United" becomes "Nvited"

Parents Day - S5-E6
Continuity mistake: In the scene where Arnold's parents leave for the last time, Grandma's hair is dark at first, then it becomes purplish-gray.

Operation Ruthless/The Vacant Lot - S1-E7
Continuity mistake: In Operation Ruthless, the pedal in Helga & Phoebe's bumper car changes color from black to gray to black between shots.

Tutoring Torvald / Gerald Comes Over - S1-E13
Continuity mistake: Arnold switches from his left hand to his right hand when taking the test.
Suggested correction: I don't believe it is ever established whether Arnold is left or right-handed in the series. Without confirmation; he may very well be part of the ambidextrous population that uses both. I used to write with both hands myself in school.
Except the mistake is saying he switches hands in between shots during the test. Not that one episode he's left handed and the next he's right handed. Yes, the teacher was shown for about 1-2 seconds before the switch happened. But these mistakes should be considered valid as an actual mistake in animation rather than to suggest to the audience that for no reason whatsoever, Arnold switches hands to finish the test, and only does so when not seen.
I do believe he is ambidextrous. In "The longest Monday" he is using chop sticks with his right hand. Many other episodes he uses his left hand to write.

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.

Love and Cheese / Weighing Harold - S4-E7
Trivia: In 'Weighing Harold', when Arnold is in the comics store, there are comics in the background referencing other Nickelodeon shows, "Catdog" and "Spongebob Squarepants." The former had aired almost exactly a year before this episode did ('Catdog' premiered on 4/4/1998 and this episode on 4/7/1999). The later aired about a month after this episode.

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).

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.

Revealing mistake: Phoebe is going into the school in one scene, but as the camera moves she vanishes while on screen.

Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: When Sid decides to be a slob again, he picks up his frog and you can see that he has gloves on. When the following shot shows a close-up of the frog in Sid's hand, the hand holding the frog does not have gloves on.

Parents Day - S5-E6
Continuity mistake: In the tricycle section of the last event, the chinpiece of Helga's helmet keeps switching colors.

The List/Haunted Train - S1-E8
Continuity mistake: At the beginning of The Haunted Train Grandpa's ice cream changes color from orange to white.

Olga Comes Home / Sally's Comet - S1-E15
Continuity mistake: In Sally's Comet when Gerald, Arnold, and Grandpa are in the traffic jam on the way to Hawk Mountain the position and type of cars around them change. When we first see the car there is a truck there and no room for them to turn around. When they do turn around then there is room and the truck is gone.

Coach Wittenberg / Four-Eyed Jack - S2-E16
Factual error: When the kids are bowling for the first time, the placement of the pins is wrong, and when Arnold is bowling there are 11 pins.

Eating Contest / Rhonda's Glasses - S2-E13
Continuity mistake: In the last scene, Rhonda is seen through the bus's rear window as sitting in the back of the bus - but just before that, she's talking to the girl with the frizzy hair in a seat closer to the front.

Freeze Frame / Phoebe Cheats - S2-E8
Continuity mistake: The next night, Phoebe's bed clothes switch back and forth between her nightgown and her dress.