These are probably the two most recognizable drum machines found in '80s music. Both came out in 1980 and would hold through most of the decade, although they peaked at different times.

THE LINN LM-1 DRUM MACHINE

http://en.wikipedia.org/wiki/Linn_LM-1

Wikipedia also provided an excellent list of Songs that have used this instrument:

Blancmange - "Happy Families"
Chemical Brothers - "It Doesn't Matter"
Prince "When Doves Cry" "1999" "Lets Go Crazy" "777-9311"
Culture Club - "Karma Chameleon", "The War Song"
Don Henley - "The Boys of Summer", "Dirty Laundry" (beginning of song)
The Human League - "Don't You Want Me", "Dare, Hysteria"
Icehouse - "Great Southern Land", "Hey Little Girl"
Gary Numan - "Dance"
George Benson - "Turn Your Love Around"
Steve Hackett - "Cured"
Elton John - "Nobody Wins"
Madonna - "Burning Up", "Everybody", "Rain"
Michael Jackson - "Thriller", "Wanna Be Startin' Something"
Queen - "I Want to Break Free", "Radio Ga-Ga"
Stevie Wonder - "Part Time Lover"
The Cars - "Drive" (possibly LinnDrum)
Peter Gabriel - "Shock the Monkey" and almost all the other songs on his fourth album.
Justin Timberlake - "Until the End of Time"
Rod Stewart - "Young Turks"
F.R. David - "Words"
Vanity - "Wild Animals"
Bill Wolfer - "Wolf"
Matthew Friedberger - "Holy Ghost Language School", "Winter Women"
Hall & Oates - "Maneater" (heard at beginning of song and during saxophone solo when acoustic drums drop out)
Ready For The World - "Oh Sheila"
Most Stock, Aitken, and Waterman productions during the 1980s (possibly LinnDrum and/or Linn 9000)
Huey Lewis and the News - "Heart of Rock & Roll"

You can tell that usage probably peaked from 1982-1985 here, and is more in tune with the "New Wave '80s".

Here are some YouTube videos that demonstrate this instrument well:

http://www.youtube.com/watch?v=lHU7tU_jHCE&feature=related

http://www.youtube.com/watch?v=z3cTT1vFTpM&feature=related(Ouch!) I can't even find a single copy available on eBay!

THE ROLAND TR-808

This is much more iconic in old school hip-hop/r&b hits, particularly in the 1986-89 period.

This time, because you probably enjoyed the LM-1, we'll do the videos first.

http://www.youtube.com/watch?v=jUx7ax62OBo

http://www.youtube.com/watch?v=sBg4h_GEhfk&feature=related

That Cowbell and Hi-Hat Combo is KILLER!

I found this to be a better source for artist information, although it doesn't pick out individual songs (Such as Marvin Gaye's Sexual Healing, the first big hit with the TR-808.
http://www.vintagesynth.com/roland/808.shtml

Oddly enough, the prices of both drum machines today aren't much different then they were in the '80s. They have stood the test of time very well. It's even more intriguing to learn about the dozens and dozens of synths/drum machines in addition to the artists/songs so you can be an expert at '80s music. It was a very tough decision in determining the more iconic of the two, but I'll have to go with the TR-808. That snare/conga/hi-hat/cowbell is so cheesy-sounding it instantly gives you the "that's '80s!" feel.

Gotta go with the 808 here, too, if for no other reason than that musicians have been singing its praises in their lyrics for two decades.

Most recently, Kanye West's album 808s & Heartbreak, released in 2008, used the 808.

Going further back, and skipping a lot, because these are just the ones that come to mind off the top of my head:

"'Cause I'll be goin' boom like an 808..."
- Blaque, 808, 1999

"I'm the TR-808, huh, but just call me D-Nice..."
- D-Nice, Call me D-Nice, 1990

"Like an 808 bass drum, we sound real good..."
- Derek B, Bad Young Brother, 1987

...and of course, 808 State, who went one step beyond singing about it or dedicating an album to it, but who named themselves after it.

Gotta go with the 808 here, too, if for no other reason than that musicians have been singing its praises in their lyrics for two decades.

Most recently, Kanye West's album 808s & Heartbreak, released in 2008, used the 808.

Going further back, and skipping a lot, because these are just the ones that come to mind off the top of my head:

"'Cause I'll be goin' boom like an 808..."
- Blaque, 808, 1999

"I'm the TR-808, huh, but just call me D-Nice..."
- D-Nice, Call me D-Nice, 1990

"Like an 808 bass drum, we sound real good..."
- Derek B, Bad Young Brother, 1987

...and of course, 808 State, who went one step beyond singing about it or dedicating an album to it, but who named themselves after it.


A couple more:

"'Cause the 808 kick drum makes the girlies get dumb..."
- Sir Mix-A-Lot, "Posse On Broadway", 1988

"But I know y'all wanted that 808, can you feel that B-A-S-S bass?..."
- Outkast, "The Way You Move", 2003

i am the owner of the tr808 roland in mint condition and i would like to exchange it against a Linn lM-1

Weeeelll, I'll tell you what.  As a listener, it all depends on whose programming it.  I always say, good music is good music whether you're using computers or a couple of spoons and a hollow log.

Roland TR 808
http://www.youtube.com/watch?v=9lDCYjb8RHk

Linn LM 1
http://www.youtube.com/watch?v=GBde1lQ_U24

I'd go with The Roland 808.