Just about every song on the Fallout 3 soundtrack could be counted as Awesome Music, but special mention goes to "I Don't Want to Set the World on Fire," especially if you take its lyrics and apply them to the events of the game.
Many of the songs from Fallout 3 are deliciously ironic. "Let's Go Sunning" comes to mind.
And for sheer Ear Worm-itude, "Butcher Pete". Which is now stuck in your head and will be for the next fifteen hours. You're welcome.
Also notable for getting stuck in your head and refusing to leave: "Bongo, bongo, bongo, I don't wanna leave the Congo, oh no no no no..."
And not just songs, soundtracks are awesome too. Take "Battle 1" playing when fighting a Super Mutant Behemoth with Fat Man.
And "Battle 5", which is actually titled "Behemoth" on the soundtrack.
From New Vegas:
Fallout: New Vegas is not without its Ear Worms. See: "Jingle Jangle Jingle", "Something's Gotta Give", and "Ain't That A Kick In the Head".
If ever there were a song to make headshots by, "Big Iron" is it.
"Lone Star" is always a good one to listen while roaming the Mojave Wasteland.
I know my home is waiting for me by the river shore! I know that all the ones I love will welcome me once more! In dreams I see them now, though it seems I'm bound to roam My thoughts are still of Texas and of Home!