Cat Stevens and Harold and Maude.
Rob Zombie did most if not all of the music for House of 1000 Corpses, as well as making the actual film, IIRC.
Neil Young wrote and performed (solo, I’m pretty sure) all the music in Jim Jarmusch’s Dead Man.
The Bad Livers’ Danny Barnes wrote and performed on all the music in Richard Linklater’s The Newton Boys, both period jazz tunes and the orchestral stuff.
The Gourds perform all the songs and incidental music in the excellent beard-growing documentary Growin’ a Beard.
I seem to remember all the music in Paris, Texas being written and performed by Ry Cooder, I could check into it – it’s hard to Google info on this, all you get is a million pages of people selling copies of the damn thing.
Not hard at all if you know where to look 
Oddly, though, there are no actual details there on the soundtrack.
True, they had the lion’s share of songs, but other groups have songs on the soundtrack: The Trammps and Yvonne Elliman come immediately to mind.
Vince Guraldi did the soundtrack and score (I believe) to A Boy Named Charlie Brown along with the first couple of television specials.
Speaking of Ry Cooder, he did the soundtrack for the movie Southern Comfort, though unfortunately they didn’t put out a soundtrack for that one. Too bad, because there’s some great eerie swamp-rock sounding stuff in that movie.
However, there is a CD of some of Cooder’s movie soundtrack music, which contains some of the music from Southern Comfort called Music by Ry Cooder. I might have to pickl that one up…
No, he didn’t . Peince did a single (won an award for it too), but Elfman was the composer.
Prince was credited with the 9 song soundtrack album (not just a single).
Danny Elfman was credited with the Original Motion Picture Score.
It occurs to me, it’d be a nice turnaround if they made a movie about Scott Joplin and had Sting do the soundtrack.
Mark Knopfler - Local Hero
Big Country - Restless Natives
Richard Thompson - Grizzlyman
Vangelis - Chariots of Fire and Blade Runner.
Not exactly - there is a medley of traditional pre-Joplin tunes (*Listen to the Mockingbird, Turkey in the Straw, * and some other that escapes my memory at the moment) in carousel-style, and there’s also one other on it that I’m sure Joplin didn’t write. I’ll have to take a look at my liner notes to that one when I get home.
I’m thinking they did the soundtrack for Legend (the fairytale/fantasy with Tom Cruise, Mia Sara, and Tim Curry).
What I meant was that there was no simple way to tell whether Cooder played all the songs (or, as I believe, all the music) that were (was) in the film (movie). As you pointed out, IMDB was mute on the topic (well, sure, I’m not a picture of clarity myself, and I haven’t seen the movie in 15 years or more), and soundtrack albums don’t always contain just songs that were in the movie, which is what I think the OP wants. I think.
Anyway, one of things I find most irksome about movies these days is how they feel the need to stop everything and hype the soundtrack album every 20 minutes or so by playing intrusive bits of what often turn out to be bad covers of good songs by a bunch of anonymous group-de-jour bands signed to the label owned by the same mega-conglomerate that owns the movie studio.
So did Britney Spears perform all the songs from her movie, or what?
Here’s a link to that Dune project…it would have been the highest geek-cred-factor movie in history 
I don’t recall if they did the entire soundtrack or not, but The Alan Parsons Project did the music for Ladyhawke. I still like the movie, but it’s hard to not be distracted by the music – it just seems jarringly out of place now.
Wang Chung did To Live and Die in LA
Phil Collins did Disney’s Tarzan