Some top choices in other people's posts here, and it's nice to see other people also like my faves, like Goin' Down and Under-Assistant West Coast Promotion Man.
A personal favorite is "Somebody's Gonna Get Their Head Kicked In Tonight" on the b-side of Fleetwood Mac's "Man of the World". "Stone Free" on the back of "Hey Joe", too.
"Gloria" by Them was originally a b-side!
An artist on top form is likely to be producing such great music that their b-sides are better than others' a-sides - The Beatles, indeed and Hendrix, and early Elvis Costello singles too.
With UK sixties singles, you occasionally get an a-side that is a commercial song, but a b-side that is the group playing an old cover-song that they used to play in the clubs pre-fame - and you get to hear great white R&B that is the reason people liked them originally. Peter Jay and the Jaywalkers, Manfred Mann, and even Billy J Kramer, singles are well worth flipping-over.
Sometimes we see "live" b-sides, which make you think that the band were so pleased with what they played, that they just "had to" get it released. REM sometimes did it, and Bowie's live "Panic In Detroit" on (was it) "Sorrow". Best-In-Show of live b-sides is probably Bob Dylan's "Just Like Tom Thumb's Blues" live in '66, on the "I Want You" single (in UK) - worth whatever you have to pay for it, and better than the version on the "Albert Hall 1966" album.