Grandpa's Packard / Phoebe's Little Problem - S5-E19
Continuity mistake: Phoebe's Little Problem: When Phoebe tells the lunch lady what she wants and she questions her choices, the lunch lady is standing behind the counter, as she should be. When she is shown again two shots later, she is standing in front of the counter, when she should be behind it. (00:04:03)

Continuity mistake: At the start, Lila is sitting on a bench near the school wall. However, in the shot where Sheena says, "Oh, that's wonderful," Lila and the bench are missing; then they reappear a moment later. (00:01:13)

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)
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)

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. This happens because the shot is taken from the episode "Helga's Love Potion."
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"

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.

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

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.
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.
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'm ambidextrous myself, and I often switched hands when taking tests. It was very useful for preventing hand cramps. It only takes a couple of seconds to switch the pencil to the other hand, so I don't see why Arnold can't do it within that span of time too.
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.
Suggested correction: I agree with the other correction, but it's also worth noting that in the production order, "Rich Kid" was made before "Arnold's Room." They were just aired out of order, just like many of the other episodes.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: After Phoebe's eyeglasses are stepped on, when she tells Helga, "Out with the old Phoebe, in with the new," Phoebe's belt buckle and two brown pouches have vanished, then reappear in the next shot. Later, while Phoebe sits on the desk during detention, her belt buckle is missing in her close-ups, and it reappears mid-shot as she crosses her legs. Then, when Phoebe's classmates share how they miss the "old" Phoebe, her buckle and the two brown pouches vanish, but they all reappear in a moment. (00:05:05)