Okay, so I did my Happy Dance on the street in front of my local PAC when I found out that Meat Loaf is going to be there in November. I freaking worship at the altar of Jim Steinman. The man is a lyrical genius to me.
I put him only one step behind Bob Dylan, who writes the best lyrics ever.
And third is Leonard Cohan.
I’m talking lyrics and poetry here, not music.
I’d put Dylan and Cohen as #'s 1 and 2, with shout outs to John Hiatt, Lyle Lovett and Steve Earle.
Joni Mitchell/Leonard Cohen tie.
Bob Hunter is up there. Bob D is my favorite. I often hear people rank Lovett & Cohen as great lyricists, but I’m in the dark. Any song recommendations?
Judging from your username, you might be inclined to include Robin Williamson & Mike Heron, as I am…
I’m also a big fan of Dylan and Cohen (and I like Steinman a lot, too). I’d add Paul Simon to my short list.
At his best, John Prine gets close or into to that second tier.
I’m not sure anyone up there (including the top two) as written anything as heartbreaking yet not cloying as “Sam Stone”, or as unexpected yet true as, say Barbara Lewis Hare Krishna Beauregard (“If heartaches were commercials, we’d all be on TV” ).
Did the OP just accidentally leave off “(besides Dylan)”, or did he forget that Dylan’s still alive?
For Cohen, I’d start with “Famous Blue Raincoat”,“Suzanne”, “Hallelujah” (link is to the Rufus Wainwright version, my personal favorite), and “Everybody Knows”.
As for my two cents, basically echoing the above - Dylan and Cohen are at the top, everyone else is harder to rank. Paul Simon is pretty high on my list, though.
Wow. I always heard that as, “If her legs were commercials, we’d all be on TV” which I thought was even more poetic. Diction was never Prine’s forte.
For lyrical poets, I prefer Morrissey, Nick Cave, Billy Bragg and Michael Gira.
I wondered if this was a common mishearing so I googled and found a page of misheard Prine lyrics. Two people had reported this exact error and one of them was me so I’ve apparently been disabused of this error before but it didn’t take. Ignorance is like a slasher-movie killer. You can kill it but as soon as you turn your back it gets back up and comes after you again. Good luck fighting it.
I absolutely do, RaVG! Robin Williamson was my very first thought. But obscurity and years of ‘dying cows/turn that twee shit off’’ derision have beaten me down. 
I nominate John Darnielle.
Dylan and Cohen are the obvious top spots, but I’m really happy to see that my other two favourites, Nick Cave and John Darnielle, have already received shoutouts. Additionally, Tom Waits has produced stuff of staggering genius, as has Shane MacGowan.
Thanks HMHW, Tom Waits gets my vote. His words can tear you up from so many directions.
Another shout for Shane MacGowan. Poguetry - top tier.
Morrissey and Mark E Smith both phenomenal - harder to judge lyricists who are in great bands, though. The music takes over a bit, something that didn’t trouble Dylan, Cohen, or MacGowan.
On a more domestic level, Nigel Blackwell from Half man Half biscuit has got one hell of a body of work behind him. They know where things are at B&Q - has anyone ever sang a more cutting line?
Lanny Poffo
Just as I’m such a huge fan, I’ll add Adam Duritz (from Counting Crows) to the list.
Ohhhh…
Ray Davies and Paul Simon.
I do not think Elvis Costello should be overlooked in this discussion.
Guy Clark is one of my favorites.