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.
Suggested correction: I agree with the other correction, but it's also worth noting that in the production order, "Arnold Saves Sid" was made before "New Teacher." They were just aired out of order, just like many of the other episodes.
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.

Freeze Frame / Phoebe Cheats - S2-E8
Continuity mistake: The next night, Phoebe's bed clothes switch back and forth between her nightgown and her dress.

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)

The High Life / Best Friends - S2-E10
Other mistake: Best Friends: When Arnold walks across the table during the fashion show, a small chunk of the spotlight vanishes mid-shot. (00:22:13)

The High Life / Best Friends - S2-E10
Continuity mistake: Best Friends: During the fashion show, in the close-up of Arnold showing off the antennas on his costume, the curtains that he came out of are missing. They are also missing in the shot after Arnold tells Gerald, "Just smile and move, okay?" (00:22:22 - 00:22:50)

The High Life / Best Friends - S2-E10
Other mistake: Best Friends: When Gerald is first revealed during the fashion show, the curtains are transparent. They are never transparent in any other shot. (00:22:26)
The High Life / Best Friends - S2-E10
Deliberate mistake: In "The High Life", there is a scene where Stinky submerges his head in a tank of water. He delivers the line, "Gee willickers, it's still ticking", and everyone somehow understands what he is saying with his head underwater.
Suggested correction: Nobody responds to what Stinky says, so there's no reason to assume that everyone understood him.
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.
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
Revealing mistake: Helga's Boyfriend: When Arnold walks up to Helga to tell her that Stinky loves her, he asks if he can talk to her about something, and in the shot of Helga, there is a line at the bottom of the screen going through her legs, cutting them off. (00:08:04)
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)

Helga's Boyfriend / Crush on Teacher - S2-E14
Other mistake: Crush on Teacher: When the kids are playing baseball and the baseball hits Arnold, as he says, "I'm fine, Miss Felter," his eyelids are not colored in. (00:13:02)

Helga's Boyfriend / Crush on Teacher - S2-E14
Other mistake: Crush on Teacher: When Arnold and Gerald are walking to school and Arnold gets distracted by the birds and flowers, as he says, "Look at the flowers," his eyelids are not colored in. (00:14:37)
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
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 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)