Alphabetically

ABBA: The Movie
The Bodyguard
Can't Stop the Music
The Commitments
Cool As Ice
Crossroads
Desperately Seeking Susan
Fear Of a Black Hat
Glitter
Graffiti Bridge
Head
I Am Trying To Break Your Heart
Jailhouse Rock
The Jazz Singer
Krush Groove
Madonna: Truth or Dare
Metallica: Some Kind of Monster
Moonwalker
Notorious
Poetic Justice
Purple Rain
Pure Country
Rock Around the Clock
The Rose
SelenaSgt.Pepper's Lonely Hearts Club Band
Sid & Nancy
Stop Making Sense
Spiceworld
That Thing You Do!
That's the Way of the World
Tommy
U2: Rattle and Hum
UHF
Under the Cherry Moon
The Wall
The Wiz

By Category

Author's Note: Obviously there are films that don't fit perfectly into just one category. I've made a judgment call on those that straddle the fence. Unlinked films are not yet posted.

The Pop Movie
These movies star a musician, feature multiple songs by that musician, and usually have some sort of autobiographical bent.
Can't Stop the Music
Cool As Ice
Glitter
Graffiti Bridge
Head
Jailhouse Rock
The Jazz Singer
Krush Groove
Moonwalker
Purple Rain
Pure Country
Spiceworld
That's the Way of the World
UHF
Under the Cherry Moon

The Starring Vehicle
These films star a musician, but may only feature one or two of their songs, or none at all.
The Bodyguard
Crossroads
Desperately Seeking Susan
Poetic Justice

The Biopic
A recreation of a musician's life.
Notorious
Selena
Sid & Nancy

The Documentary
Non-fiction representation of a musician's life using actual footage.
I Am Trying To Break Your Heart
Madonna: Truth or Dare
Metallica: Some Kind of Monster

The Pop Musical
Film featuring pop songs sung by characters in the film.
Rock Around the Clock
Sgt.Pepper's Lonely Hearts Club Band
Tommy
The Wall
The Wiz

The Concert Film
Self-explanatory, I hope!
ABBA: The Movie
Stop Making Sense
U2: Rattle and Hum

The Fake Band
A film about a band or musician who doesn't actually exist.
The Commitments
Fear Of a Black Hat
The Rose
That Thing You Do!

By Year

1956
Rock Around the Clock

1957
Jailhouse Rock

1968
Head

1975
That's the Way of the World
Tommy

1977
ABBA: The Movie

1978
Sgt.Pepper's Lonely Hearts Club Band
The Wiz

1979
The Rose

1980
Can't Stop the Music
The Jazz Singer

1982
The Wall

1984
Purple Rain
Stop Making Sense

1985
Desperately Seeking Susan
Krush Groove

1986
Sid & Nancy
Under the Cherry Moon

1988
Moonwalker
U2: Rattle and Hum

1989
UHF

1990
Graffiti Bridge

1991
The Commitments
Cool As Ice
Madonna: Truth or Dare

1992
The Bodyguard
Pure Country

1993
Poetic Justice

1994
Fear Of a Black Hat

1996
That Thing You Do!

1997
Spiceworld
Selena

2001
Glitter

2002
Crossroads
I Am Trying To Break Your Heart

2004
Metallica: Some Kind of Monster

2009
Notorious

By Artist

ABBA
ABBA: The Movie

The Beatles
Sgt.Pepper's Lonely Hearts Club Band

Bee Gees
Sgt.Pepper's Lonely Hearts Club Band

Bjork
Dancer in the Dark

Carey, Mariah
Glitter

Commitments, The
The Commitments

Diamond, Neil
The Jazz Singer

Earth, Wind, & Fire
That's the Way of the World

Frampton, Peter
Sgt.Pepper's Lonely Hearts Club Band

Haley, Bill and the Comets
Rock Around the Clock

Houston, Whitney
The Bodyguard

Jackson, Janet
Poetic Justice

Jackson, Michael
Moonwalker
The Wiz

Madonna
Desperately Seeking Susan
Madonna: Truth or Dare

Metallica
Metallica: Some Kind of Monster

Monkees, The
Head

Notorious B.IG., The
Notorious

N.W.H
Fear Of a Black Hat

Pink Floyd
The Wall

Platters, The
Rock Around the Clock

Presley, Elvis
Jailhouse Rock

Prince
Graffiti Bridge
Purple Rain
Under the Cherry Moon

Rose, The
The Rose

Ross, Diana
The Wiz

RUN D.M.C.
Krush Groove

Selena
Selena

Sex Pistols, The
Sid & Nancy

Strait, George
Pure Country

Shakur, Tupac
Poetic Justice

Sheila E.
Krush Groove

Spears, Britney
Crossroads

Spice Girls
Spiceworld

Talking Heads
Stop Making Sense

Vanilla Ice
Cool As Ice

Village People, The
Can't Stop the Music

U2
U2: Rattle and Hum

The Who
Tommy

Wilco
I Am Trying To Break Your Heart

Wonders, The
That Thing You Do!

Yankovic, "Weird Al"
UHF