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)
Weird Cousin / Baby Oskar - S4-E11
Factual error: Stephen Viksten is credited twice for voicing Oskar.
Audio problem: Right after Helga is standing behind the "ARNOLD" on the pavement, when she says "Hey Arnold", her lips move as if she's saying it slowly (this is fixed in the 4th season).
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.
Door #16 / Arnold as Cupid - S1-E10
Audio problem: At the hat store, when Arnold says "Huh? What are you, nuts?", his lips don't move - but Gerald's do.
Das Subway / Wheezin' Ed - S1-E12
Factual error: There's no way Grandma could have connected two "live" wires with her bare hands without electrocuting herself, and there's certainly no way the wires could have stayed connected.
Other mistake: In the closed-captioning, when Arnold and Gerald "touch thumbs" (and you hear that "sproing" sound), the caption says "[glass shatters]"
The Aptitude Test / Oskar Gets a Job - S3-E9
Audio problem: In the comic book shop, Gerald says "Ho. Touche, sir." but his mouth doesn't move at all.

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.
Door #16 / Arnold as Cupid - S1-E10
Factual error: Ernie removes two screws from a glass panel in the skylight, but then lifts the panel from the side that's still screwed in.
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.
Abner Come Home / The Sewer King - S1-E16
Audio problem: When Arnold is playing chess with the Sewer King, Arnold jumps across the table and knocks the King over. This sequence appears to be sped up: the toilet crashes way after it is destroyed, and the King says "Noooooooooooo," yet his mouth never moves.
Magic Show / 24 Hours to Live - S1-E18
Audio problem: Helga rides past Arnold in a bus and yells "24 hours until you die!" or something like that. Her mouth never moves- it just stays open in an evil smile.

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.
Arnold's Hat/Stoop Kid - S1-E3
Audio problem: When Harold says, "Hey, what are you, scared?" his mouth stays open for the last word. It never closes for the "d" at the end.