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.
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.
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.

Continuity mistake: In "Miss Perfect", Helga, Rhonda, Sheena, Phoebe, and Nadine are cooking up ways to make Lila's (the new girl) life miserable. In one shot, Rhonda is wearing her new blue outfit. When the scene cuts to the boys and Lila, and then back to the five girls, Rhonda is wearing her standard red sweater and black pants. For the rest of that day, she is back in her new outfit.
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)

Runaway Float / Partners - S2-E20
Other mistake: Runaway Float: When Bob asks the kids who wants to ride on the float, Arnold says to them it's great the way it is. In one shot, Iggy's hair is incorrectly coloured black instead of brown. (00:05:52)

Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: When Arnold reaches the parade float on the bike, Iggy's shirt is incorrectly coloured green instead of purple. It's also green when the float rides past Mayor Dixie. (00:09:45)

Runaway Float / Partners - S2-E20
Other mistake: Runaway Float: After Nick talks about redesigning the parade float, Bob asks the kids who wants to ride on the float. Arnold then tells them it's great the way it is, and when the kids start saying, "Me! Me!", there is an animation glitch where part of Robert's head and hands get stuck in the air for about a second. (00:06:01)
Continuity mistake: Ms. Perfect: When Lila and her father are eating dinner, Lila's spoon is in her can of beans in one shot. When the angle changes and her father says, "You sure are quiet tonight," her spoon disappears from the can of beans (the flap of her can of beans is also layered under her father's left arm), then reappears in the next shot. (00:19:05)
Other mistake: Ms. Perfect: When Mr. Simmons tells the class, "I have a note from Lila's father," his right sleeve briefly turns green. (00:19:22)
Save The Tree / New Teacher - S2-E1
Audio problem: Save the Tree: At the end, when Bob says, "Yeah, I'll just build my beeper emporium in that vacant lot down on Vine Street," his mouth doesn't move until the word "build." Arnold's mouth also doesn't move when he says, "Huh?" (00:11:27)

Other mistake: Ms. Perfect: When the girls sabotage Lila's lunch, as Helga pours out Lila's milk from her thermos, the milk crosses an invisible line and disappears. (00:16:56)
Continuity mistake: Ms. Perfect: When Lila goes to Patty's locker, the books she is holding constantly change. First, she is holding two books, then three, then the red book is suddenly on top, then the red book disappears and the rectangle on the brown book changes to white, then it goes back to yellow and the blue book gets replaced by the red one, and finally, the blue book reappears (she is also suddenly not holding the locker handle, and the way she holds the books changes). (00:15:48)

Continuity mistake: Ms. Perfect: When Patty opens her locker and the jello falls on her, her head is raised above the floor, and there is no jello on her head. In the next shot, her head is on the floor, and there is jello on her head. (00:16:24)
Audio problem: Ms. Perfect: When Lila says, "Whatever you do, don't smell that cow," the boys are heard laughing, but only Stinky is shown laughing. (00:15:14)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: In the credits, actor Elliott Gould's name is misspelled as "Elliot Gould." (00:22:59)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: In the credits, Justin Shenkarow is credited twice for voicing Harold. (00:22:59)

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)

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)