Great films where music is a central theme.

From that list, some standouts (I'm being generous with the OP's concept, especially "great film"):

8 Mile
Hairspray (the original)
Wayne's World

Films about music tend to be inextricably also about the contemporary culture.

Blade Runner.
Carmen ( Saura ).
Baraka ( documentary ).
1492:Conquest of Paradise.
A Man and a Woman.
Circle of Iron ( aka The Silent Flute ).

This thread has prompted me to get several of the movies, which I really enjoyed (The Bands Visit for one). I have to add in my own 2cents, not to add a new one to those already listed, but to reinforce the need for *anybody* interested in music to watch 'The Commitments'. Not only is the music superlative, but the story line and character development are massively captivating. I do not consider myself a movie buff by any means, but I *am* a music buff, and the *only* movie I have seen multiple, multiple times (I would guess between 50 to 100) and love it more with each successive viewing.
You have to be able to tolerate the massive amounts of profanity however, which in Dublin is used as a form of punctuation without any implied profanity

Il start light and move on to more contreversial suggestions:

Those already mentioned:
Amadeus (BTW the sound track on vinyl is pretty good too!)
The soloist (so-so, too much digital enhancement for my taste)

Now for a few new ones that come to mind:
Departures (Japanese film that won best foreign film at the 2009 academy awards) The central character is cellist that looses his job. Great music and movie.

Dancer in the dark.
Bjork! and Lars von Trier! Not for everyone, but if you like contemprary electronique music it is great.

My next suggestion is about the music, within the context of the movie, and not an endorsment of the movie itself. Just watch the prologue it's enough (if not too much) Lars Von-Trier's Antichrist. The opening seen features the Aria Lascia Ch'io Pianga, it is emotionally outstanding. Having a young son, the seen really struck a cord and it is no small part due to the music.

On a lighter note, best musical TV episode (no not Glee!) is the Simpson's episode where they go to Italy to chase Side-Show-Bob and end up at la Scala, where Krusty The Clown is performing "I Pagliacci" and finally Side-Show-Bob removes Krusty from the stage just in time to sing "Vesti la Giuba"... The show inside the show inside the show, you gotta love the Simpson's

Wow, after re-reading my post I seem to have a really grim taste in movie and music.

This thread has prompted me to get several of the movies, which I really enjoyed (The Bands Visit for one). - snip

Way cool! That's exactly why I created it - to get some good recommendations from others. This is why I requested please don't just reference generic lists of films with music. I'm really interested in your personal recommendations. Glad you liked, The Bands Visit and Commitments! Both are excellent films.

Nick_sr - Sounds like you have some crossover tastes, I mentioned the wonderful film, Departures already. Though I can only take Von Trier in small doses and all of his stuff very polarizing for myself, I have to say, that first scene in Antichrist is amazing and beautiful. It is not, however, a film with with music as any part of the storyline so not a film that fits here. I'd also warn anyone that that considers that film that it is painfully difficult/disturbing content to watch at times. I personally would not recommend it to anyone, but I can say it will stick with you. In that Dogma school of cinema Von Trier is known for, on a personal recommendation apart from this thread as it has no musical content either, The Celebration, is one of my all-time favorite films. Oh, and I did enjoy Dancer in the Dark, which does belong on this thread as music is a big part of the content - but yes, not for everyone as any of Von Triers films might be described.

Overall, everyone's been adding some great suggestions. My Netflix queue thanks you! Keep'em coming