I can't readily think of a "singer-songwriter" I'm mad keen on to be honest... bits of Elton John, David Bowie, Neil Diamond, Cat Stevens, Neil Young, Paul Simon, Neil Sedaka, Carol King, Joni Mitchell...are very good. Would struggle to sit through an entire album by any of them though. Dylan - awesome canon of work but as Todd says, invariably covered better by other artists. Great songwriter, awful singer (there's a reason that some cliches are "old chestnuts" - they've been demonstrably true to too many people for too many years to be ignored!)
Regarding the "singer" part of the equation, both John and Paul blew all those mentioned out of the water for me.
Regarding the "songwriter" element, again, Lennon and/or McCartney when on fire were better overall than any of them (including Dylan) with such astonishing breadth and style.
And yet... as solo artists ALL four Beatles slightly disappoint if I'm honest - always felt they were rarely firing on all cylinders (with the exception of a few albums) but perhaps that's because I know what they were capable of when their talents combined. I always felt each of their post-Beatle extremes would have been balanced, curbed and channelled by the input/presence of the others.
Paul was the most versatile and prolific, and the keenest live performer, he would have been up there with the best in my opinion. But he spent too much of his early solo career trying too hard to please an audience desperate for "Beatle-lite" product in the vacuum of the 70s. And when I say "solo" career it's ironic how he - arguably with the greatest all round capability of the four - needed the supportive cocoon of a group around him for a full decade after the split.
John was such a magnetic, eccentric, unpredictably gifted individual (and a better musician than sometimes credited) I think he'd have been a big success in the singer-songwriter mould. Conversely to Paul, he didn't care enough about his listening public though, preferring to eschew commercial/accessible radio-friendly hits (which he could've written standing on his head) for ever more personal material which sometimes hit the mark perfectly but often went way wide. John wasn't the lazy Beatle, he was the restless one; when he got bored of "Beatles", pop records or even fame he just jettisoned them and moved on and if you didn't like it, tough!
George wasn't in the same league as Lennon or McCartney as a vocalist. For a time he could more than hold his own as a songwriter (though never on the scale of consistent quality achievable by those two) yet even more than John, he wanted to use his fame and success as a stepping stone to abandon populism (for much of the 70s at least) and urgently preach his own philosophy to the listener. Very chequered output then; at his best, a credible singer/songwriter but a mostly reluctant performer not interested enough in wanting to be one to be honest.
Ringo personified the bonhomie which helps to carry a live show and whilst lacking the talent of, say, Stephen Stills, Eric Clapton or James Taylor, had more "presence" and star quality than the three of them put together. Very limited both as a singer and a songwriter, he relied on a web of support and if it wasn't for the fact he was a Beatle - his passport to immortality - wouldn't seriously figure in the debate alongside those artists already mentioned. Still made some belting records though!!!!