It's established that the Fallout universe diverged from our own around the 1950's, and at least American culture fixated on the idea of the 'Nuclear Age'. Feasibly, music would have continued in the same vein, stagnating and reproducing the same old melodies.
I always thought it was a little depressing that almost 200 years after the war, there's hardly any sense of creativity out in the Wastes. Granted, most people are just trying to survive, but even in the richer, better protected settlements (Vault City, Rivet City, etc), no-one has the wherewithal to bang a couple of old tin cans together, let alone string out a tune.
As a personal long-term dream, I'd love, no, pay to see some burgeoning cultural growth in the Fallout universe. Music, writing, art, it can't stay buried forever. Just a couple of ideas off' the top of my head:
* A Super-Mutant writes a book for kids, 'Grog Says Safe', where Grog teaches children about the dangers of the wastes by illustrating just what happens when you run into a Deathclaw.
* A retired Raider, looking for some caps without having to spill too much blood, happens upon a battered, cracked old electric guitar. Somehow, it still works, but the notes sound so much more...savage than before.
* All over the ruins of the Wastes, bright, beautiful murals start springing up, portraits, landscapes, painted and sprayed for the world to see. An anonymous artist has found his calling.