
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.

Best Man / Cool Party - S3-E12
Other mistake: In the scene where Arnold meets the gang in the alley way, they give ideas to crash Rhonda's party. When Curly tells his idea to go to the zoo and free all the animals and there are shots of the gang looking at him confused,in two of the scenes, you'll notice that when Stinky and someone else looks at him, the background is at the school, not the alley.

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.
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.

Helga Blabs It All / Harold the Butcher - S3-E1
Continuity mistake: Helga Blabs It All: After Helga successfully snatches the message tape, when she rushes out of Sunset Arms the front door is missing its handles, but one handle reappears when Campfire Lass and other girls chase after Helga.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: When Principal Wartz tells Helga she cannot volunteer again, he removes the "revered badge" and its black crossbody strap from Helga. Helga then selects Phoebe as the next hall monitor. When Phoebe joins them on the stage, Helga is again wearing the crossbody strap with the badge in the wide shot, but it's gone in the next shot as Wartz places it on Phoebe. (00:02:10)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: While Helga is hall monitor, her watch repeatedly disappears and reappears on her wrist.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: At the start of the school assembly, when Principal Wartz thanks Helga for her month as hall monitor, Wartz's right arm is atop the lectern, with his fingers over the lectern's front edge. As Wartz gestures toward Helga with his right arm and hand, the fingers of his right hand oddly remain over the lectern's front edge. Then, after Helga announces she may volunteer again, the audience groans, and in the next shot facing Wartz, he has four hands at the lectern's edge. (00:01:35 - 00:02:00)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: During detention, the class helps Phoebe realise how much she has changed. When Helga wipes the window so Phoebe can see her reflection, in the first shot, Phoebe's reflection is not backwards as it should be. (00:10:45)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: While Phoebe is hall monitor, her watch repeatedly disappears and reappears on her wrist, even mid-shot. (00:05:00)

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)

Sid the Vampire Slayer / Big Sis - S5-E3
Other mistake: Sid the Vampire Slayer: While Sid stands in front of the mirror and points his rubber sword at Arnold and Stinky, the two ghoulish vampires, Arnold asks him, "How can we be vampires if we have reflections?" So when Sid is aware of his mistake, Arnold's eyes disappear and reappear in his reflection mid-shot.

Helga Blabs It All / Harold the Butcher - S3-E1
Continuity mistake: Helga Blabs It All: When the boys get ice cream from the Jolly Olly ice cream truck, Arnold gets a chocolate popsicle, and Gerald gets an ice cream cone. Then, when Helga falls into Arnold's attic bedroom, all the boys stare at her, and as Helga backs out of the room, the ice cream cone Gerald holds changes to a chocolate popsicle.

Save The Tree / New Teacher - S2-E1
Continuity mistake: New Teacher: Arnold's classroom door vanishes from its door frame in various shots, then the door reappears in its proper place. That door's doorknob also switches between the right side of the door and the left side a few times.

Arnold Betrays Iggy / Helga and the Nanny - S3-E10
Continuity mistake: Arnold Betrays Iggy: When Arnold goes to Iggy's home near the start, the front doorknob is on the left side of the door, but the next time Arnold goes to see Iggy to bring him the box of chocolates, the front door opens the opposite way, with its doorknob now on the right side of the door. (00:01:20 - 00:06:40)

Parents Day - S5-E6
Continuity mistake: During the flashback, when little Arnold says goodbye to his parents, in the next shot of Stella and Miles, the bag's strap is slung over his dad's left shoulder as a crossbody, but mid-shot, the strap switches to his right shoulder with the bag on the opposite side. (00:14:30)

Parents Day - S5-E6
Continuity mistake: When Grandpa enters and exits Arnold's bedroom, the door opens the opposite way in various shots, with the doorknob on either the left or right. Additionally, the light switch beside the door appears and disappears, between shots.

Helga Blabs It All / Harold the Butcher - S3-E1
Continuity mistake: Helga Blabs It All: While Helga hides in the overhead ventilation shaft, the vent cover has either four, five, or six slat openings in various shots.

Helga Blabs It All / Harold the Butcher - S3-E1
Continuity mistake: Helga Blabs It All: When Helga falls from the vent opening, she knocks the telephone and answering machine onto the floor. Just as Helga notices the tape in the answering machine, the push button telephone has changed to a phone with a rotary dial. It then changes back to push button. (00:10:05)