Jack Jones is among the singers included on a list of approved music for the town. In 1958, Jones was still an unknown. He didn't make his recording debut until 1959 and didn't achieve widespread recognition until 1962, when "Lollipops and Roses" hit the charts.
The Dave Brubeck Quartet track, Take 5, (which is played during the diner scene where Bud is asked how he knew about fire) was released in 1959. The next track that is played is Miles Davis's So What was recorded in 1959 a year after the action is meant to be taking place.
When David/Bud is pulling up to his home with the firemen to put out the burning tree, he is heard yelling, "Alright, stop...stop!" At this point you can already see his face in profile and he is clearly not saying anything.
When a rock is thrown through the window of the soda shop, it goes through the face of the woman in the painting. A moment later, another item is thrown, also breaking the window in a different place, but the woman's face is intact again.
When the soda shop is wrecked and it is discovered that the juke box still works one of the boys unplugs it after Buddy Holly's "Rave On" starts. When Bud says it's okay and plugs the juke box back in the song starts again, which would not have happened with a circa 1958 juke box. It would have started up where the needle was left by the unplugging.
The flag outside the school has 48 stars, the correct number for 1958. The one atop the fire truck has 50 stars. By 1958, it was well known that the 50-star flag would be introduced soon, and 50-star flags were available, but it still seems unlikely that an official vehicle would be flying an unofficial flag (especially since the official flag in 1959 had 48 stars until July 3rd, and 49 stars from July 4th until the end of the year).
The jukebox in the diner is unplugged while Buddy Holly's song is playing, yet when the jukebox is plugged back in, the song starts from the beginning instead of slowly starting up from the point at which the machine was unplugged.
On their way to school on the first day, Jennifer/Mary Sue gets angry and pulls her hair clips out - just before Skip pulls up in his car. After Skip drives away her hair is suddenly clipped back again.
When Bud is about to go to work for the first time, Mary Sue complains to him about having to wear falsies on her date. When she's talking to her date at the restaurant, the falsies are gone. She's wearing them again right after her date is over and she walks into the house.
In the bowling alley sequence, the scene begins with two or three 7-10 splits being picked up. Later in the same scene when the mayor is speaking with the scores behind him there are no 8 pin spares listed, only 9 pin spares.
When Jennifer and David are fighting over the remote control before the Pleasantville Marathon starts at 6:30, they momentarily go to the Prevue Channel. The time shown on that channel is 1:16, not 6:30.
The TV repairman leaves right before the beginning of the marathon and is seen smiling in his van, looking back at the house. The same shot is seen again at the end of the movie, despite the narrator of the marathon indicating the end of its first hour. It is unlikely the repairman would have simply sat there for a whole hour.
In the bowling alley, we are shown a sequence of 7-10 splits converted to spares. This should be marked as an 8-/ on a scorecard. However on the scoreboard projected behind Bob, there are no frames with any 8s. Even if the scores hadn't been recorded yet, we should at the very least see 8s with the top right squares blank.
When Bud pulls up in the fire truck to put out the tree fire, he pulls a charged hose from the back of the truck. While some hoses in the back of an engine are connected to the engine's pump, water would not flow until the pump was engaged.
When the TV repairman talks to David and Jennifer from the television after they've just been transported to Pleasantville, behind him, the test pattern's position on the screen is slightly different in the close-ups. This was done deliberately "to indicate Knott's changing attitudes towards Bud and Mary Sue".
Betty Parker's eyes are obviously green, and her hair is reddish, when she transitions to color. This is very evident when Budd helps her conceal her skin color with makeup. While her husband and the mayor may have simply not noticed, in later scenes her eyes are again gray, and her hair almost black, until the scene in which Bill Johnson removes it in the diner. Note that the makeup powder could not be used on the eyes, or on the hair.
At the very end of the movie, George Parker and Betty Parker are sitting on a bench. The camera pans from George to Betty, then back to where George was sitting, and he "turns into" Mr. Johnson. If you look at Betty before the camera goes back to Mr. Johnson, you can see her bounce. This is from Mr. Johnson sitting down in George's place.
The goof item below may give away important plot points.
Several things go unexplained at the end of the film. The TV repairman disappears for most of the third act (save for a brief appearance at the end where he smiles and drives off), and it is never disclosed how David figures out how to transport himself back to the real world, or why the TV repairman now seems to be fine with Pleasantville having been quite drastically altered, after having been so passionately opposed to the changes being made. Also, while Jennifer decides to stay in Pleasantville as Mary Sue (pointing out that she likely would not have been admitted to any college in real life and wants something better for herself), it's never clarified if the "real" Bud returns to Pleasantville upon David's departure. It is also rather peculiar how Margaret and Betty appear to have been told the truth about David's real identity, but don't seem fazed that they are witnessing his "trip" into another realm, just sad that he's leaving. Conversely, neither the real Mary Sue nor Bud are shown to be transported to the real world upon David and Jennifer's arrival into the television world at the beginning of the film. If Jennifer had stayed in Pleasantville, wouldn't her mother back in the real world miss her or at least wonder where she is? The film ends without David or his mother addressing Jennifer's disappearance, and without anyone addressing what had happened to the Parker children or where they are.