Louisville Post-Rock
Hardcore kids who learned to play quiet: clean guitars in odd meters, spoken vocals, basements in Kentucky.
What you're really asking
You're asking about a family tree, not a genre. One small city, maybe forty teenagers, who played in each other's hardcore bands from 1985 on and then slowed everything down — clean guitars, odd meters, half-spoken vocals, the loud part withheld until it hurts.
Every name here appears on somebody else's record.
Where it sits
Louisville, mid-80s. Squirrel Bait — teenagers including David Grubbs, Brian McMahan, Britt Walford — put out two records on Homestead and broke up by 1988.
Grubbs went to Bastro, then Gastr del Sol. McMahan and Walford formed Slint, cut Tweez with Steve Albini, then Spiderland for Touch and Go in 1991.
That record redirected a decade. Rodan followed on Chicago's Quarterstick in 1994, and when they split the pieces went everywhere: Shipping News formed in autumn 1996 when Rodan's Jason Noble and Jeff Mueller wrote music for the NPR program This American Life; Mueller also co-founded June of 44, and their debut came out on Quarterstick in June 1995.
Tara Jane O'Neil went solo. Noble built Rachel's as a chamber group.
Touch and Go/Quarterstick carried most of it; Initial Records held the local hardcore end.
The scene
Who: Brian McMahan, Britt Walford, David Pajo, Jason Noble, Jeff Mueller, Tara Jane O'Neil, Kevin Coultas, David Grubbs, Will Oldham around the edges. Rooms over actual venues — basements, the Rocket House, Lucky Anchor.
What: two guitars in competing tunings playing interlocking parts rather than chords, bass carrying melody, drums that stop the song dead. Vocals spoken, muttered, then screamed once.
Songs run eight to fourteen minutes and withhold the payoff. How: nobody auditioned.
You got in by already being in somebody's band, and by recording with Albini, Bob Weston or Brian Paulson, who kept the room sound in.
The leading records

Slint
Spiderland
1991
Six songs, Chicago, August 1990 with Brian Paulson — the template for quiet-then-detonate everyone copied after.

Rodan
Rusty
1994
Noble, Mueller, O'Neil and Coultas; "The Everyday World of Bodies" runs past fourteen minutes and never settles.

June of 44
Engine Takes to the Water
1995
Cut at Plantain Recording House, Brooklyn, December 1994, recorded by James Murphy — years before LCD Soundsystem.

Shipping News
Very Soon, and in Pleasant Company
2001
Recorded at Lucky Anchor in Louisville across April–May 2000; the sleeve credits a sweater as an instrument.