Following in the format of my FMA Anime-1 Music Master List, here is the corresponding list for FMA: Brotherhood!
There are three complete soundtracks containing the background music from the TV series. Note that these technically have the same titles as the OSTs from Anime-1, but are in fact different.
"Fullmetal Alchemist Original Soundtrack 1"
"Fullmetal Alchemist Original Soundtrack 2"
"Fullmetal Alchemist Original Soundtrack 3"
The "Final Best" Set compiles all of the full-length opening and ending themes.
"Fullmetal Alchemist Final Best"
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. I have provided links to FMA-themed Limited Editions where possible, but some of these are already out of print. There are still regular versions available elsewhere.
TV Series OP/ED Singles:
1st OP "Again" by Yui
1st OP "Uso (Lie)" by Sid
2nd OP "Holgram" by Nico Touches the Walls
2nd ED "Let It Out" by Miho Fukuhara
It's worth noting there is a slightly different version of "Let It Out" (tamer guitar sequence in the middle) on Miho's full-length album "Music is My Life"
3rd OP "Golden Time Lover" by Sukima Switch
3rd ED "Tsunaida Te (Hands Held)" by Lil'B
4th OP "Period" by Chemistry
4th ED "Shunkan Sentimental (Sentimental Moment)" by Scandal
5th OP "Rain" by Sid
5th ED "Ray of Light" by Shoko Nakagawa
The song "Drama" by Sid was used in the Wii game Fullmetal Alchemist: Prince of the Dawn" and can be found on the band's album "Hikari." "Uso" also appears on this album.
There were several character singles released, as well as a compilation album. The single versions contain instrumental versions of some of the songs.
"THE ALCHEMISTS featuring Edward Elric"
"THE ALCHEMISTS featuring Alphonse Elric"
"Theme of Ling Yao by THE ALCHEMISTS"
"Theme of Lan Fan by THE ALCHEMISTS"
"Theme of FULLMETAL ALCHEMIST by The Alchemists"
Remixes, Covers, Random
The Exit Trance series has produced several remixes of OP/ED themes:
"Again" appears on EXIT TRANCE PRESENTS Umauma Dekiru Trance wo Tsukuttemita 8 Hansei ....emasu
"Hologram," "Shunkan Sentimental," and "Ray of Light" all appear on EXIT TRANCE PRESENTS Umauma Dekiru Trance wo Tsukuttemita 9 Koko ni Byoin wo Tateyo
"Golden Time Lover" appears on EXIT TRANCE PRESENTS CODE SPEED アニメ トランス BEST 10 (sorry, couldn't find a proper link for this one!)
Not an "Official" dub or remix, per se, but I wanted to add Y. Chang's English version of "Golden Time Lover" to this list, because I think it's pretty great!