There’s an old Style Council Mini LP (remember them?) that has a gorgeous acoustic version of “Headstart for Happiness.” Paul Weller is one of my idols, and this version made me learn guitar just so I could play it.
How about Moonshiner by Uncle Tupelo?
Brilliant and haunting. Can you say Amen?!
Old Man and Needle and the Damage Done by Neil Young
Diary by Bread (stop snickering, it’s kinda poignant a capella)
Guinevere by Crosby, Stills & Nash, also Wooden Ships
Taxi by Harry Chapin
Waiting For My Real Life To Begin - Colin Hay
Winter and Closer, by Josh Radin. You can listen to a couple of samples on his site (including those two). The current version of Closer appears to have a little more instrumentation (sounds like a synthesizer doing chords, mostly) than the original version I heard, but it’s mostly him and guitar.
Hello - Evanescence. This one is primarily vocals and piano. Although there’s a bridge section with what sounds like strings (since it’s a break from the vocals).
Hallelujah by Leonard Cohen, Sweet Baby James by James Taylor. And most of S&G songs are guitar friendly. My personal favorite is American Tune.
Beck- Lost Cause
Frank Sinatra (written by Rodgers and Hart) - My Funny Valentine
Nina Simone- Sinnerman
Also, Nobody’s Fault, from Mutations. And Jackass from *Odelay. *
God, Beck is awesome.
Nothing like any of the songs in the OP, but may I recommend:
“Lost in the Stars”, written by Kurt Weill and Maxwell Anderson. I am particularly pleased by Wesla Whitfield’s rendition for heartbreaking poignance.
Now, I am the compleat atheist, but this song makes me understand why many people might long for the solace that trust and belief in a god can bring.
Dr Hook—“Queen of the Silver Dollar”
Harry Chapin
—“Cat’s in the Cradle”
—“Shooting Star”
—“Six-String Orchestra”
Willy Nelson—“Pancho and Lefty”
“Sloop John B.” Traditional, famously covered by the Beach Boys.
I don’t know the title or the original artist, but it’s a standard in the coffee houses.
Most of the soundtrack from Oh, Brother, Where Art Thou?
Most of the soundtrack from Forrest Gump.
Good one. I once heard a geeky white guy sing it with an acoustic guitar, and it was still great!
This kid doesn’t fit in the “classic” folk description.
But he does it well and “live” is unbelievable.
Also, you can hear him in the chorus -he says, “Uru Cha-Cha”.
I thought it was fitting for “ZebraShaSha”
Elliott Smith had a number in this category. Some highlights include “Needle In The Hay”, “Angeles”, “Miss Misery”, “The Biggest Lie”, “Say Yes”, “Between the Bars”…
Simon and Garfunkel can probably fill a whole page. “Bleecker Street”, “Wednesday Morning 3 AM”, “April Come She Will”, “Flowers Never Bend With The Rainfall”, “Sounds of Silence” (acoustic version), etc.
Johnny Cash’s “Hurt” also pops to mind here…
To the OP, nice seeing another Silvio Rodriguez fan!
One man and his guitar, you say? Possibly with some backing instruments? You want Bards, Barking Division, m’lad.
Or otherwise known as Billy Bragg
Some of my favourites:
Valentine’s Day Is Over
Must I Paint You A Picture
Waiting For The Great Leap Forward
Levi Stubbs’ Tears
Greetings To The New Brunette
I’ll put forth Van Diemen’s Land on U2’s “Rattle and Hum.” It’s just The Edge playing his guitar and singing a beautiful tune. Really great.
Unfortunately, U2 hasn’t performed this in over a decade 
I was going to mention this one.
“The Wind” - Cat Stevens
“Stand By Me” must have been mentioned already, but I missed it if so.
“A Reminder” - Radiohead
“Hallelujah” - Leonard Cohen originally, Jeff Buckley’s cover is the best, as everyone knows
“Arms of the Angel” and “When She Loved Me” - Sarah McLachlin
“May It Be” - Enya
“Pie Jesu” - Andrew Lloyd Weber (sung by Sarah Brightman and a boy soprano in the original “Requiem” album).
All haunting, spare and powerful.