Night Music

"Sunday Night, later named Michelob Presents Night Music, was an NBC late-night television show which aired for two seasons between 1988 and 1990 as a showcase for jazz and eclectic musical artists. ... " (Quote from the Wikipedia entry.)

"A revelation to music fans, this show appeared and disappeared with little fanfare, being light-years ahead of the bulk of it's audience in style and substance. Every possible realm of music was boldly explored...from minimalist art-noise to traditional folk styles and everything beyond and in-between. Absolute genius, a marvelously noncommercial experiment that was doomed to failure despite itself. ..." (Quote from user comment at the IMDb entry for 'Sunday Night')

The show was hosted by Jools Holland (only first season) and David Sanborn.
Now, see, this is music television that doesn't suck.
I've collected some YouTube clips from episodes of the tv show:

-Milton Nascimento - "Canções e Momentos"
-Milton Nascimento - "Cravo e Canela"
-Milton Nascimento & Naná Vasconcelos - "San Vicente"

-Dizzy Gillespie - "Tin Tin Deo"
-Dizzy Gillespie - "Ooh-Shoo-Be-Doo-Bee"

-Al Green - "The Message Is Love"

Al Green.

-Curtis Mayfield - "It's All Right"
-Curtis Mayfield - "Pusherman"
-Curtis Mayfield (with Taylor Dane) - "People Get Ready"

-Youssou N'Dour - "My Daughter"

-Marcus Roberts - "Blue Monk"

-Phillip Bailey - "Fascination"
-Phillip Bailey - "I Thought I'd Never See the Day"

-Ambitious Lovers - "Admit It"
-Ambitious Lovers - "Copy Me"

-David Lindley - "Bon Temps Roulez"
-David Lindley - "Werewolves of London"

-Taylor Dane - "Where Were You When I Needed You?"

-Shinehead

-Sonny Rollins

-Leonard Cohen - "Tower Of Song"
-Leonard Cohen & Sonny Rollins - "Who By Fire"

-Ken Nordine

-Was (Not Was) - "Can't Turn You Loose"
-Was (Not Was) - "Earth to Doris/Robot Girl"
-Was (Not Was) - "Hello Operator"
-Was (Not Was) with Frank Sinatra Jr. - "Wedding Vows in Vegas"

-John Lurie and Naná Vasconcelos

-Stevie Ray Vaughan - "Crossfire"

-Maria McKee - "Breath"

-Debbie Harry - "Calamari"

-Pharoah Sanders - "Thembi"

-Pops Staples - "Nobody's Fault But Mine"

-Pere Ubu - "Breath"

-Ivo Papazov and his Wedding Band

-Pretty Phat - "Pretty Phat Girls"

-Pixies - "Monkey Gone To Heaven" /"Tame"

-Sun Ra Arkestra - "Face the Music"
-Sun Ra Arkestra - "Retrospect" (starting with Sun Ra being briefly interviewed by Sanborn):

Video added by ekp34

-Sting (with Bill Frisell and Fareed Haque) - "Ain't No Sunshine"

-Fareed Haque (with Sting)

-Bill Frisell and Band

-Carla Thomas - "Gee Whiz (Look At His Eyes)"

-Rufus Thomas - "Walking the Dog"
-Rufus Thomas - "The Funky Chicken"

-Mary Margaret O'Hara

Sting does "Why Does the Sun Shine?"

-Zahar (with Hassan Hakmoun)

-Djavan (Note: Video problem towards the end)

-Eric Clapton - "Hard Times"

-Robert Cray & Eric Clapton - "Old Love"

-Sonic Youth - "Silver Rocket"

-Diamanda Galas

-The Residents - "Teddy Bear"
-The Residents - "From The Plains To Mexico"

-Conway Twitty & The Residents - "When You're Cool"
-Conway Twitty - "It's Only Make Believe"

-Aster Aweke
-More with Aster Aweke

-The Kronos Quartet - "Spoonfull"
-The Kronos Quartet - "Four, for Tango"
-The Kronos Quartet - "Foxy Lady"

-Red Hot Chili Peppers - "Subway to Venus"
-Red Hot Chili Peppers - "Sexy Mexican Maid"

-Toots Thielemans - "Sophisticated Lady"
-Toots Thielemans - "Little Suede Shoes"
-'Toots Thielemans plays the harmonica with a turkey baster'

-Charlie Haden & his Liberation Orchestra - "Spiritual"
-Charlie Haden & his Liberation Orchestra - "The Anthem of the African National Congress"

-Nick Cave - "The Mercy Seat"
-Nick Cave, Charlie Haden and Toots Thielemans - "Hey Joe"

-Annabouboula

-Graham Parker - "Slash & Burn"
-Graham Parker - "My Love's Strong"

-NRBQ with Phil Woods - "Little Floater"

-Abbey Lincoln - "First Song"
-Abbey Lincoln and Phil Woods - "High Fly"

-Steve Turre and his Sea Shells

Steve Turre.

-Phil Woods

-Lakim Shabazz and his D.J.

'NRBQ Blows up a New Kids on the Block Album'

-Bob Weir & Rob Wasserman - "The Victim"

-Artis (Spoon Man)

-Richard Thompson - "Jerusalem on the Jukebox"
-Richard Thompson with Jo-El Sonnier - "Tearstained letter"

-Tim Berne - "Hong Kong Sad Song / More Coffee"

-Jo-el Sonnier

-John Cale (& B.J. Cole)
-John Cale, Shawn Colvin & Richard Thompson - "Heartbreak Hotel"

-Shawn Colvin

-Hank Crawford - "Don't Cry Baby"
-Hank Crawford - "The Peeper"

-Miles Davis - "Hannibal"
-Miles Davis - "Tutu"

-The Modern Jazz Quartet
-The Modern Jazz Quartet - "The Legendary Profile"

The Modern Jazz Quartet

-Lounge Lizards - "Voice Of Chunk"
-Lounge Lizards - "Uncle Jerry"

-NRBQ - "I Got a Rocket in my Pocket" :

Video clip added by scmm42

-NRBQ - "Wild Weekend"

-Warren Zevon - "Splendid Isolation"

-Dan Hicks - "I Scare Myself"

-Screamin' Jay Hawkins - "I Put A Spell On You"
-Screamin' Jay Hawkins - "Old Man River"
-Screamin' Jay Hawkins, Bob Weir, Bongwater and the Fabulous Pussywillows

Note: Thanks to bobby4000 and especially to scmm42 for uploading these great clips! Thanks also to the other uploaders of course.
Some tracks are missing track titles. If you know a title or two please leave a comment.
And note that the quality for some of the video clips is not super good.