Girl Trouble / School Dance - S3-E18
Continuity mistake: Besides the fact that when Helga is sharpening her pencil, the desk arrangement changes, one of the students visible when she looks at Arnold is...Helga.
Longest Monday / Eugene's Pet - S2-E4
Continuity mistake: In "Longest Monday" at one point in the episode, Arnold and Gerald run off of a bus and down an alley. Two fifth graders come chasing after them. In the next scene, you see Arnold and Gerald hiding in the sewer under the 5th graders. Arnold then crawls through a very narrow tube, and Gerald follows. You then see the end of the tube and Gerald come out of the it first, and Arnold climbs out last. It would be impossible for Arnold and Gerald to switch places inside the tube because it was way too narrow.

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)
Continuity mistake: When the four boys form their line (just before the adults call out "Arnold"), watch Harold's tooth; it goes from the right side of his mouth (as you look at him) to his left side.

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.
Continuity mistake: When Lila is finishing her first joke, Rhonda suddenly switches from her French outfit to her "usual" one; also, during the Jell-O incident, her beret disappears and then reappears.

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.
Helga's Makeover/The Old Building - S1-E4
Continuity mistake: When the classroom is first shown, the first two desks against the wall near the door are occupied by Curly and Eugene, Gerald and Arnold are sitting where Curly and Eugene were shown.

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.
Tutoring Torvald / Gerald Comes Over - S1-E13
Continuity mistake: Arnold and Gerald have to climb stairs to get to Oskar's apartment, but this would mean the building has three floors, and the outside views indicate that there are only two (plus the attic).
Continuity mistake: The first time you see the line of girls, Rhonda's shoes are "low-cut", but just before Helga says "Move it, football head", the shoes suddenly extend much higher (like they normally do in the stories).
Crabby Author / Rich Kid - S3-E17
Continuity mistake: In "Rich Kid" Lorenzo is introduced as a new student, but he was in a previous episode titled "Arnold's Room."
Suggested correction: Other than the few episodes where a specific time of year is taking place; these episodes are not meant to be taken to be shown in chronological order. Just roughly in the same 4th grade school year.
Arnold's Hat/Stoop Kid - S1-E3
Continuity mistake: When Grandpa is telling Arnold about how how he tried to cover his head when his hair falls out, the camera cuts back to him with his arm on Arnold's shoulder and Arnold has his hat on, even though the whole story is about him losing his hat.
Tour de Pond / Teachers Strike - S2-E18
Continuity mistake: The kids' teacher in "Teachers Strike" is Miss Slovak, who was their teacher in the first season, but in the first episode of this season they had a new teacher, Mr. Simmons.
Suggested correction: Other than the few episodes that take place during a specific time of year; these episodes are not meant to be taken in chronological order. Just roughly in the same 4th grade school year.
Arnold Saves Sid / Hookey - S2-E7
Continuity mistake: The kids' teacher in this episode is Miss Slovak, who was their teacher in the first season, but in the first episode of this season they have a new teacher, Mr. Simmons.
Suggested correction: Other than the few episodes that take place during a specific time of year; these episodes are not meant to be taken in chronological order. Just roughly in the same 4th grade school year.
Arnold's Thanksgiving - S3-E14
Continuity mistake: When Joey is shown in close-up, his missing tooth looks like it's there but it's covered with something black and his missing tooth keeps reappearing and disappearing in the last scenes.
Arnold Saves Sid / Hookey - S2-E7
Continuity mistake: The clock in the police station looks like it says it's after 5:00, but school is in session.
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.

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.