Lots of good picks so far. If I had to only recommend one music related documentary, it would probably be "Muscle Shoals."
Here's a few that have not been mentioned yet:
BADFINGER - Director's Cut
BECOMING LED ZEPPELIN - If you can see (and hear) this in an IMAX theater, it is well worth the time and effort.
HUNG UP ON A DREAM (The Zombies)
IMMEDIATE FAMILY - Directed by Damon Tedesco ("The Wrecking Crew") about the careers of studio musicians: Danny Kortchmar - Russ Kunkel - Lee Sklar & Waddy Wachtel.
SEARCHING FOR SUGARMAN (Rodriguez)