Deliberate mistake: When Evan/August first meets Arthur at the park, after playing, you see Arthur leaving his spot without taking the cable and (presumably) the amp as well.
Deliberate mistake: Throughout the film, scenes of several Cole Porter productions are shown being performed. These scenes all contain a well-known musical number. Two scenes in particular (after the Alanis Morissette number and immediately after the "So in Love" duet) end with the lead singer/singers bowing to the audience and then waving the other cast members on stage to take a company bow. The problem is that neither of those numbers appear at the end of the shows being "performed." The "So in Love" duet, for example, most definitely appears in the musical "Kiss Me, Kate" well before the Act Two finale. I assume that these mistakes were filmed deliberately to emphasize the importance of the songs in question, but they are still errors.
Deliberate mistake: During "Izzy"'s inaugural concert with Steel Dragon, there are two shots that are recycled during the first song. One is the rear shot of the guitarist in a low stance rocking back and forth, and the other is Jennifer Aniston raising her arms in the air and screaming for the band. Not only is the repeat a mistake, but when they reuse the shot of the guitarist, the second time it is not in time with the music.
Deliberate mistake: Before going on stage in the infamous purple coat for the first time in the movie, the Kid is in his dressing room. The silver jewelry/decoration on his coat is on the left shoulder. In all subsequent shots, the silver jewelry/decoration is on his right shoulder.
Deliberate mistake: The same group of people wearing the same clothes, including a dancer and bartender, are shown sitting around a bar on two different disco nights. On the first disco night we see them when Tony leaves the bar to go outside after being informed by Joey, what Double J is doing in the car. We see these people again at the start of the second disco night, both nights when we see them, the song "If I Can't Have You" is being played. (00:21:20 - 00:56:30)
Deliberate mistake: The song "If you give me your attention" (from "Princess Ida") and the Mikado's song (from "The Mikado," of course) are performed with their central verses omitted. This was no doubt to save time and, in the case of the Mikado's song, to avoid inclusion of the line "is blacked like a ni**er" (altered in 1948 to "is painted with vigour").
Deliberate mistake: In the housing block that Dave's house is in, the house numbers do not match - because In one shot of Claire walking past Dave's house, you can see that the house numbers are in the 1500's - one homes number is clearly 1516. However Dave's house number is 1958. However this is a deliberate mistake as the 1958 on Dave's house is a reference to when the Chipmunk fever was born. (00:23:50 - 00:32:35)
Deliberate mistake: Directly after Nicola has her baby, she sits up and holds a baby that is clean, umbilical cord-free, and has teeth. There is no possible way this could happen.
Deliberate mistake: When Wayne begins playing the "Excalibur" guitar, the clerk stops him and directs him to a "No Stairway to Heaven" sign. The problem is that the guitar riff Wayne started playing sounded nothing like "Stairway to Heaven" so the clerk had no reason to interrupt him. Evidently the intro to "Stairway" was actually played in the theatrical version of the film, but its exclusion in the home video release makes the scene rather dubious.