I reserve the right to edit this list as any overlooked songs come to my attention. As it stands, I've done my best to not only compile existing songs but also to provide links to singles or albums where possible.
I see that Tombow started a similar list a couple of years ago, but never finished it. This one is in the same spirit, but in a briefer format for quick viewing and finding!
There are three complete soundtracks containing the background music from the TV series. There is one from the movie.
"Fullmetal Alchemist Original Soundtrack 1"
"Fullmetal Alchemist Original Soundtrack 2"
"Fullmetal Alchemist Original Soundtrack 3"
"Fullmetal Alchemist: The Movie - Conqueror of Shamballa Original Soundtrack"
The "Complete Best" Set compiles all of the full-length opening and ending themes.
"Fullmetal Alchemist Complete Best"
There is also an instrumental album compiling full-length orchestral versions of all of the themes.
"Fullmetal Alchemist - The Instrumentals"
Most of the CD singles from the individual bands or singers contain instrumental/karaoke and/or TV size versions of the opening and ending themes.
TV Series OP/ED Singles:
1st OP ”Melissa" by Porno Grafitti
1st OP "Kesenai Tsumi (Indelible Sin)" by Nana Kitade
1st OP has an acoustic version as well: "Kesenai Tsumi (Indelible Sin) - raw 'breath' track" Version
2nd OP "Ready Steady Go" by L'Arc-en-Ciel
2nd ED "Tobira no Mukou He (To the Other Side of the Door)" by Yellow Generation (no link, sorry!)
3rd OP "Undo" by Cool Joke
3rd ED "Motherland" by Crystal Kay
4th OP "Rewrite" by Asian Kung-Fu Generation
4th ED "I Will" by Sowelu
Movie OP/ED Singles
"Link" by L'Arc-en-Ciel
"Lost Heaven" does not appear on any soundtrack or single, but can be found on L'Arc-en-Ciel's full-length album "Awake."
There was a soundtrack released for the second PS2 game. It is mostly very video-game-esque BG music, along with a few "normal" songs: "Flower of Heart" by Motoko Kumai, "Emotionally" by Saori Yamada, and the previously mentioned 'raw' version of "Kesenai Tsumi."
"Fullmetal Alchemist: Tobenai Tenshi (The Broken Angel)"
In addition to the regular OP and ED themes, the 3rd PS2 game featured a second single from Nana Kitade.
"Kanashimi no Kizu" by Nana Kitade
There were several character singles released, as well as a 2-disc compilation album. The single versions contain instrumental versions of some of the songs.
"Hagaren Song File - Edward Elric"
"Hagaren Song File - Alphonse Elric"
"Hagaren Song File - Roy Mustang"
"Hagaren Song File - Winry Rockbell"
"Hagaren Song File - Maes Hughes"
"Hagaren Song File - Best Compilation"
Remixes, Covers, Random
Nana Kitade's Singles Compilation album features a remix of "Kesenai Tsumi" called the "~I Scream~ Tempura" Version.
"Berry Berry Singles" by Nana Kitade
Yellow Generation's full-length studio album features a remix of "To the Other Side of the Door" called the "FULLMETAL Mix."
"life-sized portrait" by Yellow Generation
Edward Elric seiyuu Romi Paku/Park recorded a jazzy version of "To the Other Side of the Door" on her full-length album.
"Toi Kioku" by Romi Paku
Dub actor Vic Mignogna recorded an English version of "Brothers."
Vic also composed an original "inspired by FMA"-type song, "Nothing I Won't Give," which can be found on his album.
"Metafiction" by Vic Mignogna
There is also a remix album featuring remixes of his versions of "Brothers" and "Nothing I Won't Give."
"Revix" by Vic Mignogna
Spanish J-Pop cover artist Salome Anjari recorded a cover of "Kesenai Tsumi."
Chinese singer Eason Chen recorded a Cantonese version of "Melissa."
Cover artist Piano Squall recorded a nice piano version of "Bratja (Brothers)."
The Exit Trance series features a techno remix of "Link" on its' movie album.
Exit Trance Presents SPEED Anime Trance Best The Movie 2
NEW! As of today, 3/28/12, Cool Joke just released a new CD single of "Undo" called "Undo -8 Years Later-," featuring a new version of the song called "Undo - Wild Side Ver.-"