I've been holding this one for awhile, but why wait any longer?
"Torn and Frayed" from Exile on Main Street (1972) is one of the Stones most successful forays into country music. But it's not standard honky tonk, owing as much to Memphis soul, and even gospel, as it does Bakersfield.
Many speculate that Gram Parsons who was present at the French chateau where Exile was recorded, had a hand in this one, and it certainly recalls some of his best work. But the song was written by Jagger-Richards and the only credited guest musicians are Al Perkins on pedal steel guitar, Nicky Hopkins on piano and Jim Price playing organ. But many things were in the air during the recording of Exile and Gram's country-rock was among them.
The Stones haven't performed this one a lot live, but here's a great one from 2002: