OP here. Thanks for all the responses so far. 
And thanks for providing so many YouTube links so I can find the songs easily.
Keep 'em coming.
My “favorite song that no one has heard” is What Goes On by The Velvet Underground. I love the way it progresses. It starts off somewhat benign, and then the keyboards come in. An ear-bleeding guitar solo takes it over the top. Pure joy.
Awesome. Have you heard the version on the “1969” live album?
Lou and Doug probably had wrist injuries after that
They’re probably not “obscure”, but some years ago I had a cassette tape of ‘hits of the 80’s’ and three songs were brand new to me. “The Politics of Dancing” by Re-Flex, “The Wild Wild West” by Escape Club, and “Heartbreak Beat” by the Psychedelic Furs. Never heard any of them, anywhere, in years previous and when I did, couldn’t stop playing that tape.
“Wild Wild West” was a top-10 hit, I believe.
My pic is by L7, “Pretend We’re Dead.” Not really obscure, but I have never heard in on the radio, and I’ve lived in five different states over the last 15 years, on both coasts and the midwest.
The Super Session version of Bob Dylan’s “It takes a lot to laugh, it takes a train to cry.”
Mike Bloomfield, Al Kooper and Steven Stills.
That album also has an excellent version of “Season of the Witch” mentioned above by Rick.
Those were all hits in the 80’s. “Heartbreak Beat” is probably the lesser known of the three. I was a Psychedelic Furs fan and they had several really good songs back then.
Led Zepplin’s Trampled Underfoot.
Springsteen’s Does this bus stop at 82nd street?.
It’s from his very first album. I heard him perform it a few years ago in a concert, and was delighted.
I think we can agree that it is not really all that obscure if a song is a regular part of the set lists of the School of Rock and is often played by 7 year old kids.
“Thank Christ for the Bomb” by The Groundhogs (proto-metal, play very loud).
“Snow in San Anselmo” by Van Morrison (Van the man, with a choir, channels William Wordsworth in a California exurb).
“7 and 7 is” by Love (pure, angry punk from a psychedelic band from 10 years before punk).
“Gala” by The Waterboys. (Another one to play very loud. Although the wind down, and, arguably, the intro too, goes on too long, the central song is intense and amazing.)
That’s a good song. I used to hear it on the college radio station back in the day.
Can’t believe I, of all people, forgot about this one: “Near Wild Heaven” by R.E.M.
Robin Trower in general qualifies but Day Of The Eagle in particular:
Shoot to Kill, by 1994
Sterling probably did too.
Broadcast seem a bit under-rated to me.
Papercuts:
“Stop It”-Pylon
I’ve got to add a couple more of my favorites. Most of the songs by Brent Best of Slobberbone are classic songs waiting to become popular. I have no idea how they aren’t stars. Maybe it’s the name. Either way, here’s two of my favorites from them (it’s country rock, that still counts as rock, right?):
Pinball Song
I Can Tell Your Love Was Waining
And I love this thread. Even if the song you post is one I end up not liking, I don’t know that yet. More! More!