Steely Phil / Quantity Time - S2-E12
Continuity mistake: Chinese Checkers is supposed to be with ten pieces per side, not six.

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.
Helga's Boyfriend / Crush on Teacher - S2-E14
Continuity mistake: Crush on Teacher: When the kids are playing baseball and Miss Felter drives by, Harold starts walking to the side of the road, and Helga is standing on the road. In the next shot, Harold is already standing to the side of the road, and Helga is standing on the sidewalk. Helga's mask also disappears. (00:12:42)

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 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)
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)
Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: In the first and last overhead shots of the detention classroom, the "fire" sign, the fire extinguisher, and light switch are not on the wall beside the door, though all are on the wall in other shots during this scene. (00:09:30)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Harold's Bar Mitzvah: In the beginning, when Stinky asks Harold if he wants to play stickball, there is no wristband on his arm. Harold says he has to go do his lessons, and in the next shot, a wristband appears on Stinky's arm, then disappears. Later on, when Harold comes out of the synagogue, his wristbands are back on his arms. (00:12:03)
Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: During the school assembly, Principal Wartz's necktie has diagonal stripes, and its top knot stripes switch to the opposite direction, then revert to the proper direction.

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.
Coach Wittenberg / Four-Eyed Jack - S2-E16
Continuity mistake: Coach Wittenburg's anchor tattoo changes color from red to purple in between shots.
Suggested correction: No, his tattoo is always red.
Tour de Pond / Teachers Strike - S2-E18
Continuity mistake: Tour de Pond: When Arnold makes a boat out of a block of wood, there is a blue cup on the newspapers with gray scissors and another orange object in it. In the next shot, the cup becomes green, and the scissors and other object become pink. As the scene continues, they change colours a few more times. The pencils also become the same colour as the newspapers. (00:02:11 - 00:03:49)

Tour de Pond / Teachers Strike - S2-E18
Continuity mistake: Tour de Pond: When Gerald puts the boat made out of the block of wood in the fish tank, the sail is pointing towards the right, but when the boat sinks, it's pointing towards the left. (00:02:18)

Tour de Pond / Teachers Strike - S2-E18
Continuity mistake: Tour de Pond: When Rex notices that Arnold's boat is winning and starts running, the red and yellow squares on his grandfather's badge change to red and black. (00:10:18)

Tour de Pond / Teachers Strike - S2-E18
Continuity mistake: Tour de Pond: When Rex's grandfather walks up to Arnold's Grandpa at the end, his badge has eight squares on it, but in the next shot, it changes to four stripes. (00:11:20)
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.
Suggested correction: I agree with the other correction, but it's also worth noting that in the production order, "Teacher's Strike" was made before "New Teacher" was. They were just aired out of order, just like many of the other episodes.

Part-Time Friends / Bio Square - S2-E19
Continuity mistake: Part-Time Friends: When Arnold and Gerald get part-time jobs at the flower shop, while Mrs. Vitello explains things to them, she asks if she's going too fast, and in the next close-up, Arnold's apron vanishes, then it reappears.

Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: When Arnold first sees the beeper float, he says, "How can they control that thing?", and Gerald is sitting on the curb to the right of Arnold. In the next shot, Gerald is suddenly behind him. Arnold is also a lot further away from the curb than he originally was. (00:08:16)
Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: The big beeper float constantly changes appearance during the parade. Some examples: The sign saying "Let's Go Downtown" on the front changes to "Let' Go Downtown" in one shot. The sign saying "Let's Go Downtown" on the back is missing in one shot. The orange base is sometimes blue. The blue city design around the base sometimes isn't there, and it even changes colour. (00:08:17)