During the 1964-1965 school year, Mr. Holland helps his class appreciate Bach by showing the connection between a Bach composition and "A Lover's Concerto" by the Toys. The record came out in fall 1965, during the next school year.
If Mr. Holland is 60 in 1995, he would be 30 in 1965, when he tells Iris how he fell in love with John Coltrane's music in 1950, when he was 15. Coltrane's first session as a leader was on 31 May 1957, so he couldn't have had a John Coltrane record then.
In a montage of mid-late Seventies musical events seen just prior to scenes set in 1980, David Byrne of Talking Heads is seen performing in the over-sized Big Suit designed especially for the band's concert film Stop Making Sense, released in 1984.
In the scene where Mr. Holland is starting a Marching Band for the first time, a saxophonist is using a Rovner ligature on his mouthpiece, instead of a typical metal one. The Grant High School Band from Portland, Oregon was used in the filming, and the student used his personal horn/mouthpiece and ligature, not realizing that it didn't exist in the time period they were filming. Rovner ligatures weren't invented until the mid 70's, and definitely didn't become well-used by musicians until the late 80's.
In front of the drug store waiting for the midnight bus - the bus pulls away and we get a clear shot of the crosswalk. In the 1960s when the movie was set, crosswalk patterns were lines across the street from corner to corner, as opposed to the modern way which is what we see there.
Throughout the film, it is obvious that the band "musicians" (particularly the horns and woodwinds) aren't really playing their instruments. Beyond the bizarre fingerings, great sound pours out, yet the players' lips, cheeks, and neck muscles show no movement or exertion.
When young Gertrude Lang showed up late, Mr. Holland mentions that she left her clarinet there the other day. Then he asks her to play for him. She would not be able to just pick up the clarinet and play. The reed would be dry and hard. To play the clarinet, Gertrude would've needed to get the reed soft and wet.
During Lou's funeral two of the rifle party (a master sergeant and a buck sergeant) have their unit patches (Sixth Army) on the wrong shoulder. It should be on the left. The right should is where their combat patch would be (I think they may have had unit patches on both shoulders). And BTW: The "Present Arms" command mentioned in another comment was meant to go with the following playing of Taps. But because the two scenes are run together, you hear it as a command to the the rifle squad directly after the rifle volley.
Shortly after Mr. Holland's son opens a window, causing papers to fly all over the room, a medium shot shows a young man sitting in a chair. He stands up. When the camera cuts to a long shot, he's back in the chair, and stands up again.
Mr. Holland plays Lovers Concerto to his class and asks who wrote It. A student replies, 'The Toys.' Mr Holland corrects him and says "That was Minuet in G, by Johan Sebastian Bach." The piece is often attributed to Bach as it was found in a book kept by Bach's wife. It was in fact written by Christian Petzold (1677-1733).
After finding out his son is deaf, Mr. Holland teaches his class about Beethoven's deafness while writing his seventh symphony. Beethoven lost his hearing gradually. He wasn't completely deaf until he wrote his ninth symphony.
In several scenes, Mr. Holland conducts with his baton in his left hand. Traditional conductors generally conduct with their batons in their right hand, even if they're left-handed, due to centuries of tradition. It's the same reason left-handed violinists don't hold the bow in their left hand.
In the early months of 1965, the faculty meets to decide what new textbooks will be used the following semester (1965-66) - but even that long ago, decisions about which texts to use would have been made a year or more in advance to facilitate board approval, budgets, etc.
In the Gershwin revue audition scene, Mr. Holland is genuinely surprised and impressed by Rowena's talent and asks her name. However, in the later diner scene, Rowena explains that she loved being in his classes and in the scene where Iris sees her name in the program, she had been in the earlier school production of "My Fair Lady." It is highly unlikely then, that Mr. Holland would have no idea who she is.