Obscure bands

Bdp, Wild Turkey is a good one as well Klaatu and Can added by others. And Box of Frogs as well.
There's a few on the list that are great but I wouldn't consider obscure though.

More entries for the list (quotes are from All Music); please forgive where not truly obscure but more like forgotten or under-recognized):

- Frozen Ghost ("every track is rock-solid and probably gone forever, further adding to the mystique of one of the all-time should-have-beens.").
- Died Pretty
- The Adventures ("Soothing harmonies, ringing guitars, and lush synths ... shimmering, hummable pop").
- The Silencers ("consistently melodic and occasionally breathtaking...unfailingly pleasant and sometimes inspired...the band's consistency went unrewarded and ... was largely ignored").
- Propaganda
- Concrete Blonde (yes, they did have hit MTV videos).
- Willi Jones ("Almost every song is a winner ... why wasn't there at least one more album?")
- Bill Lloyd ("essential addition to the collection of any fan of melodic, unflashy, guitar-based power pop").
- The Records ("undeniably made some of the best singles of the [70's] era...definitive proof of the band's generally overlooked brilliance. Anyone interested in power-pop should start here").
- Antena ("chanteuse-whisper vocals provide one of the clues to the group's uniqueness, as do the pencil-thin synth lines and genteel bossa nova drum machine patterns...the wonder of this rediscovery").
- Zero 7 ("chiseling textured aural atmospheres...accomplished slice of soulful genius").
- Air ("electronica ... ideas wrapped up in music that was engaging, warm, and irresistible").
- Art of Noise 
- Juno Reactor ("Art of Noise for the 21st century coupled with atmospheric merging with world music elements from all over the planet...mammoth sounds at hand -- deep, dark moods, moments of widescreen cinema").
- Run Lola Run movie soundtrack ("fast-paced techno...one of the most cohesive soundtrack albums in recent memory, and the music is excellent").
- Twin Peaks ("borders on fever-dream jazz...dark, cloying, and obsessive -- and one of the best scores ever written for television").
- Donnie Iris (maybe not so obscure and too pop for some, but nevertheless a fun rocker with great song hooks and harmonies; per Allmusic "Donnie Iris is one of the classiest rock singers who deserved widespread stardom but never achieved it".)

Bdp,

I couldn't help chuckling a bit as I read your most recent post.  It almost reads like a scene from "Spinal Tap".  That must have been a fun time and it sounds like a few of those guys have had a decent run in the music making business, which is always good to hear.

And also need to add the following as groups who had major hits but have body of work that can be considered obscure to many (quotes from AllMusic):

- The Church (considered a one-hit wonder in the US ("Under the Milky Way") that nevertheless recorded 19 albums and still going strong; per AllMusic "jangling guitar pop with opaque wordplay to create a lush, melancholy brand of neo-psych rich in texture and melody"). 
- Jesus Jones (were on top of the world but fell hard with no one buying later albums that still contained catchy songs).
- Inspiral Carpets ("released some of the best singles of the '90s").
- The Outfield ("The world quit listening, but The Outfield continued to make tight, glistening pop records...sharp harmonies, crackling guitar, and the always remarkable vocals...The Outfield recedes into history as yet another shining power pop band unjustly ignored and lost forever").
- The Ocean Blue ("melodic guitar and synth-driven pop that ranks among the best ... indisputably well-crafted albums").

’Few more:
Kasenatz-Katz Singing Orchestral Circus (Quick Joey Small) 
Slaughter & The Dogs (Quick Joey Small)
Fever Tree (60s Acid band)
Simply Red (English torch
Stories (Brother Louie)
The Shoes (used name before Beatles could co-opt!)
The Left Banke (Pretty Ballerina -very trippy)