Maybe some movies that were later adapted into more traditional musicals, like the original Hairspray or Footloose? posted by sparrow89 at 5:09 AM on May 21, 2012
I can think of three examples for different ways of looking at this question. All or none of them might be what you're after:
The Life Aquatic with Steve Zissou has, in the film but not directly related to the plot, Seu Jorge singing David Bowie songs in Portuguese. (Here's Life on Mars, and the album on Spotify.)
Much of Magnolia is based around Aimee Mann songs, including the famous Wise Up montage. The songs were not written for the film but feature heavily. (Spotify.)
Eddie Vedder wrote music for Into the Wild, the songs for which are featured in the film but not performed or sung by any characters. (Guaranteed won a Golden Globe; Spotify again.)
I'll throw out Amadeus and 8 Mile as occupying opposite ends of the non-showtunes musical film spectrum. posted by drlith at 5:47 AM on May 21, 2012 [6 favorites]
This is Spinal Tap
Any of the Beatles' movies posted by Dolley at 5:50 AM on May 21, 2012 [1 favorite]
I always thought Hustle and Flow was, in it's own weird way, a musical. There is definitely music, and it is definitely not showtunes-style. posted by windbox at 6:13 AM on May 21, 2012 [1 favorite]
South Park: Bigger, Longer, Uncut was kind of a musical. Maybe more so than you are looking for though. posted by ArgentCorvid at 6:16 AM on May 21, 2012
Hmm. On the face of it, The Singing Detective looks like the "show tunes out of nowhere" kind of show you're trying to avoid, but it's really very, very different; it embraces the fact that "people bursting out of song out of nowhere" is a surreal thing and works that surreality into the plot (the main character is a little nuts, and the people-bursting-into-song is a sign that you're seeing things from his own hallucinatory perspective).
All That Jazz is similarly "show tunes out of nowhere but in a weird funky experimental-reality way". posted by EmpressCallipygos at 6:46 AM on May 21, 2012
The Big Lebowski came to mind. Also The Bodyguard. posted by daikaisho at 7:02 AM on May 21, 2012
Songcatcher is about a musicologist living in and learning about the culture and music of ~1910 Appalachia, so there's a lot of music in the movie. posted by aimedwander at 7:04 AM on May 21, 2012
The Kids Are Alright posted by chazlarson at 8:22 AM on May 21, 2012
Still Crazy is about a rock band reforming and going on tour, so there are various songs and performances. posted by shelleycat at 8:25 AM on May 21, 2012
Monsoon Wedding posted by Mchelly at 8:44 AM on May 21, 2012
Harold & Maude. The Graduate. posted by thejoshu at 8:54 AM on May 21, 2012
Cabaret? Rather than the songs coming out of nowhere, there is a rationale for them as they are part of the action - songs sung at the Kit Kat Klub. posted by paduasoy at 12:41 PM on May 21, 2012
Seconding Once. It's exactly what you're looking for. posted by cnc at 12:49 PM on May 21, 2012
If you're into flamenco at all, both Carmen and Vengo are exactly what you're looking for. posted by rjs at 1:48 PM on May 21, 2012
posted by Pomo at 5:03 AM on May 21, 2012