Not a band, either, but Elvis Costello has excellent lyrics most of the time.
Fish-era Marillion.
I think that the current rock band with the best lyrics is The Decemberists. The Tain should be proof of this alone but pretty much every one of their songs is rich with historical references, eccentric and obscure vocabulary, poetic turns of phrase, and all-around elegant rhymes that also often have a sense of humor.
Radiohead’s lyrics tend to linger with me.
The Kinks.
Crass
Dead Kennedys
They Might Be Giants
The Arrogant Worms
Most of the songs I remember for the lyrics are more lighthearted stuff you see from folks like TMBG, Dan Bern, Jill Sobule, Joss Whedon, and the kind of stuff you hear on Dr Demento.
Steely Dan and Elton John (via Bernie Taupin) have gotta get a mention in here.
Pearl Jam
Belle & Sebastian
Rock:
The Smiths and Morrissey
Counting Crows
Radiohead
U2
Nine Inch Nails
Hip Hop / Rap:
Beastie Boys
KRS-One
Jay Z
Fabolous
Kanye West
Drake
Absolutely.
I will add
Queen
Don McClean
Warren Zevon
Paul Simon
The PAINTER Francis Bacon? :dubious:
Heh, if you’re going to bring up Cop Shoot Cop I feel the need to mention Jim Thirwell of Foetus.
‘Baptised in the Hudson river, Hepatitus saves us all.’
Heh, I agree with you on all of these except Nine Inch Nails which all sound like they were written by a 15 year old Goth. I like NIN better than any of the other bands but the lyrics are pretty stupid overall.
I think Blue Oyster Cult has some interesting songs (lyrics-wise).
Clutch. Pure brilliance.
Surprised we’ve gotten this far without someone mentioning The Cure. Disintegration (btw, the best album ever) has some of the best lyrics I’ve ever read.
I’ve often wanted to start a thread on my favorite Mike Patton (Faith No More, Tomahawk, peeping Tom, etc) lyrics. He’s a master-class vocalist and a very clever lyricist.
So is Billy Bragg. Well, a socialist cause 
Greg Cartwright (Reigning Sound, Compulsive Gamblers, The Oblivians) is a master of the breakup song.
I like the oblique lyrics of early R.E.M.
The Kinks. Ray Davies isn’t a great poet (in the sense that his lyrics aren’t particularly poetic), but he’s a wonderful versifier. His songs are well-crafted, interesting, and substantial. A couple of random examples: She Bought A Hat Like Princess Marina, Celluloid Heroes
They Might Be Giants. Yeah, their lyrics can be silly, geeky, or stream-of-consciousness, but they’re clever, interesting, and at least as substantial as much of what passes for pop music. Examples: Turn Around, “Road Movie To Berlin”
Daniel Amos. The most creative band, both musically and lyrically, to come out of the Christian music industry. Songwriter Terry Taylor is one of the greatest songwriters of the rock era. Some of his songs, like The Organ Bar, I still don’t have entirely figured out, even though I find them both fascinating and moving.
Something Fierce. This Minneapolis-area band had its career cut short in the early nineties when singer/songwriter Jeff Carpenter was involved in a serious car accident, but not before recording some of the cleverest, funniest, hookiest, funnest power pop I’ve ever heard. For example, there’s “Watergate,” which uses the downfall of Nixon as an extended metaphor for a relationship gone sour: