This is a good idea for a thread but I've been reluctant to post because I just don't know what counts or does not count as obscure to people I've never met. For instance, Big Star has been an important part of my life since the early 1980s when I was in college, but I've never even heard of your other two picks. I thought of posting a list that included the Louvin Brothers, Os Mutantes, and the Vulgar Boatmen, but are they really obscure? Depending on age, location, and musical taste, no. So I wasn't sure.
That said, what the hell, I'll give it a go.
1) Judee Sill, Judee Sill. Sill was the first musician David Geffen signed when he started Asylum Records in 1971. She made two brilliant records for Asylum that went nowhere and she disappeared, dying young at age 35 from a drug overdose. She was the very definition of a cult artist until her records were reissued on CD. Her LPs only sold in the thousands in the 1970s and were hard if not impossible to find until the internet came along. Now she's much less obscure but I bet many of you are unfamiliar with her.
If so, I recommend a listen to her first, eponymous release (though her best song, the astonishing "The Kiss," is on her second LP). Sill switches from folky acoustic guitar to gospel-tinged piano playing and sings in a plaintive but appealing voice that sounds very southern Californian. Lyrically, her songs have a preoccupation with a mystical, quasi-Christian yearning for salvation (that for me avoids cliché) and has aged better, in my mind, than Robert Plant's Tolkien-inspired mysticism in the early years of Led Zeppelin. Hey, it was the early 1970s.
Musically, Sill was an intriguing mix of folk, gospel, and Bach. I've never tired of listening to her music. The round at the end of "The Lamb Ran Away with the Crown" rivals the one at the end of the Beach Boys' "God Only Knows." That's high praise.
2) And now for something completely different: Kanda Bongo Man's 1993 Soukous in Central Park. This is African Soukous music at its most joyous and mesmerizing. I'd describe Soukous (which began in the 1970s in what is now the Democratic Republic of Congo) as what happens when west Africans hear Cuban rumba music on the radio and play their version of it using electric instruments and west African poly-rhythms. It was hugely popular in the 1980s and 1990s in west and central Africa and France. Kanda Bongo Man is not among the most heralded Soukous singers and band leaders, but this live recording of his is touched by magic. Buoyant, exuberant, and infectious from beginning to end. If you get the Soukous bug listening to this, check out the music of Tabu Ley, especially the later recordings after 1970. He was the Soukous master.
Oh, and if you're a fan of the electric guitar, you really ought to listen to what Soukous guitarists do with the instrument: cascading notes played quickly but cleanly and smoothly, simultaneously intricate and melodically memorable.
3) Jason Molina recorded under various names (Jason Molina, Songs: Ohia, the Magnolia Electric Company) but all of his records ultimately revolve around his songs and his haunting voice. He sang in a midwestern (U.S.) twangy tenor that communicates heartache and vulnerability as well as any singer I can think of. (In fact, his singing reminds me a little of Alex Chilton's singing when he was in Big Star, and who has sung with more vulnerability in their voice than Chilton on Big Star's Third?) Molina's songs are usually chordally very simple (all he needed were two or three chords), but I find almost all of his songs melodically gripping. I'm no music composition theorist, but I suspect his melodic sense was more modal than modern scale-oriented. Whatever the case, I find his voice and melodic sensibility spellbinding. Lyrically, he was preoccupied with loneliness and pain (returning again and again to lightning, ghost, and bird imagery). He was no happy camper, so anyone who doesn't care for downbeat lyrics should keep a wide berth.
His most accessible record is probably The Magnolia Electric Co. from 2003. It's a boisterous Americana-esque production recorded with a studio full of sympathetic musicians. Check out the first track, "Farewell Transmission," and the last one, "Hold On Magnolia."
Personally, my favorite Molina records are Ghost Tropic and The Lioness. They're stripped down affairs, often featuring just Molina and a guitar or two, with minimal drums.
Well, that's my contribution.