Bob Schneider is one of my favorites currently. He also does some, as he says, “nasty songs”.
I think of Elvis Costello as the Cole Porter of the eighties.
More recently, Jason Mraz puts together some clever lyrics.
Paul Westerberg of The Replacements is still one of my favorites.
“Swingin’ Party”
“Here Comes a Regular”
“Can’t Hardly Wait”
“I’ll Be You”
“Achin’ to Be”
“Nobody”
“Sadly Beautiful”
I like James McMurtry
You’d a Thought (Leonard Cohen Must Die)
Too Long in the Wasteland
and Patty Griffin
Mary
Every Little Bit
Stephen Sondheim is still alive, and at 86, is still writing.
Are Rice and Webber still going?
Rice is currently working on the new Beauty and the Beast movie; Webber just finished work on the School of Rock musical.
John Darnielle, of the Mountain Goats fame, is I think one of the strongest lyricists at the moment.
Mark “Stew” Stewart, who sometimes recorded as leader of The Negro Problem, has been compared to Cole Porter. He might have peaked a few years ago, though. I’d recommend the CD Welcome Black, though it’s far from “current.”
Then there are the dozen or so truly brilliant rap/hip-hop artists – wordsmiths like Chino XL, who also was peaking over a decade ago. We’ve had threads on these sorts of “clever rap” artists, or you could dive into the wonderful and serious tome The Anthology of Rap, published about five years ago.
My top five:
Bob Dylan
Leonard Cohen
Jim Steinman
Kinky Friedman
Don McLean
About ten years ago, Paste Magazine published a list of the “100 Best Living Songwriters.” Alas, a few on the list are no longer living; and not all great songwriters are great lyricists of the sort the OP is looking for; but some certainly are, so the list may give you a few new ideas.
I remember that when the list originally appeared in the magazine, it appeared along with a mention of Mark Heard as an example of a great songwriter who is, sadly, no longer living.
Also, for anyone interested in songwriters and songwriting, there’s a generous collection of fascinating interviews with a wide variety of songwriters here.
Sara Bareilles.
Ryan Adams, Lucinda Williams, Lyle Lovett, and Tom Waits are my favorite lyricists working today.