Downtown as Fruits / Eugene's Bike - S1-E1
Continuity mistake: Downtown as Fruits: After Arnold and Gerald arrive at the school, it cuts to Helga crying about her play being ruined. When Phoebe taps her, her left arm is coming from the side of her cheese costume, but when the angle changes, it's coming from the front of her costume. (00:10:43)
Downtown as Fruits / Eugene's Bike - S1-E1
Continuity mistake: In the Eugene's Bike episode, when Arnold has to tie his shoe, he sets his books down on a bike in a bike rack. This causes the bike to tip over and makes all the other bikes in the rack to fall down. Eugene's bike falls into the middle of the street and get ran over. When Eugene finds out his bike is destroyed, he looks at Arnold and yells. Right after this you get a good look at the bike rack. You'll notice there are no bikes in or around the bike rack.
The Little Pink Book/Field Trip - S1-E2
Other mistake: Field Trip: When Lockjaw goes back into the water after Harold throws his ice cream at him, there is a shot of Arnold looking sadly in Lockjaw's direction, and he is behind the wall like he should be. Two shots later, he is mistakenly placed in front of the wall instead of behind it. (00:15:28)
The Little Pink Book/Field Trip - S1-E2
Other mistake: Field Trip: When Arnold is telling Grandma about Lockjaw, right before Grandma gets up from the couch, her left foot is meant to be on the floor, but it is instead overlapping the couch. Then, when she marches up to the couch while saying, "Like he's some sort of common criminal," her right foot overlaps the couch, and Arnold is kneeling on the backrest of the couch instead of the seat as he should be. (00:18:01)
The Little Pink Book/Field Trip - S1-E2
Other mistake: Field Trip: When Grandma uses the grappling hook to climb the wall, she falls to the other side, and she clips through the grappling hook. (00:19:17)
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"
The Little Pink Book/Field Trip - S1-E2
Continuity mistake: Arnold is talking to his Grandma about the turtle. There is a table in front of them with three items on it: a remote, a magazine, and what looks like food with a cup. In one shot, the items move around at least twice, depending on which angle the shot is.
The Little Pink Book/Field Trip - S1-E2
Continuity mistake: The orange juice container on the table jumps around when Arnold's grandpa is talking to him about his crush. First it's in the middle of the table, then it moves right. Then back and right again.
The Little Pink Book/Field Trip - S1-E2
Continuity mistake: Arnold puts the pink book on top of a bookshelf at an angle. When Helga comes into his room moments later, the book is upright.
The Little Pink Book/Field Trip - S1-E2
Other mistake: When Arnold is reading the diary at the school the other kids ask him to read more. He says "OK, but it is the last page." There are more pages. When he starts to read, then it is the last page.
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.
Arnold's Hat/Stoop Kid - S1-E3
Audio problem: When Harold says, "Hey, what are you, scared?" his mouth stays open for the last word. It never closes for the "d" at the end.
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.
6th Grade Girls/The Baseball - S1-E5
Other mistake: Sixth Grade Girls: When Arnold and Gerald are dancing with Connie and Maria, in one shot, Maria's left foot is incorrectly overlapping the right foot of the girl behind her. (00:07:32)
6th Grade Girls/The Baseball - S1-E5
Continuity mistake: While walking down the stairs, Maria only has three toes on each foot; she has four at the pool.
Other mistake: Snow: After Grandpa turns the street into an ice rink, it cuts to Arnold in the backyard (by the garage) of Sunset Arms where he's been stowing logs in a storage container. When Grandpa shows up in the backyard and tells Arnold, "You got no play ethic," we see the fronts of Green Meats and Art's Shoe Repair behind Grandpa, but that's impossible because Sunset Arms is on the same side of the street as those two stores; we should only see the buildings/stores that are across the street from the boarding house. (00:21:30)
Operation Ruthless/The Vacant Lot - S1-E7
Continuity mistake: Operation Ruthless: When the Cheese Festival is first shown, Arnold and Gerald are standing at the bumper car ride, but in the next shot, they are suddenly standing in a different location, which seems to be at the green building from the beginning of the episode. (00:03:12)
Operation Ruthless/The Vacant Lot - S1-E7
Audio problem: Operation Ruthless: When Arnold first starts repeating "Hi Ruth," the shot cuts to Gerald as he says, "Stay cool, Arnold." Arnold can still briefly be heard saying his line, but his mouth doesn't move. (00:03:42)
Operation Ruthless/The Vacant Lot - S1-E7
Revealing mistake: Operation Ruthless: When Helga and Phoebe are chasing Ruth in the bumper car, when Ruth's car stops and Helga looks back at her, Helga's right lower leg is incorrectly layered over the bumper car seat, and Phoebe's legs are colored black. (00:08:11)
Operation Ruthless/The Vacant Lot - S1-E7
Continuity mistake: Operation Ruthless: When Helga and Phoebe's bumper car flies into the air and lands on the hot dog truck, as the people start to run away, watch for one person that disappears from the shot, then reappears, and his shirt changes from green to orange. (00:08:36)
Answer: The show is based in present day times. Plus most cars aren't old fashtioned, I recall seeing a limo and not to mention a cherry picker, etc.