FWIW, Cash did not write “Sue.”
A bar fight? Cash, no doubt about it. Sinatra got punched by the manager of a casino in Las Vegas and lost a tooth. His response? “I learned never to fight a Jew in the desert!” 
(And no, for the benefit of any who may not be aware of it, he was not anti-Semetic. After all, Sammy Davis, Jr. was one of his best friends.) 
But seriously, he had many, many friends who were Jewish.
As a singer? Sinatra was in a league of his own. The art of music, technically, is extremely complex and multi-faceted if you know what you’re doing. People who know far more about music than most, who are steeped in it from childhood and who perform it at the highest levels of sophistication, are almost universally in awe of Sinatra. Not only for the way he sung, but the way he knew *how * to sing. Sinatra was the real deal. No shortcuts, nothing phony or artificial. He was without peer, and those in the know musically know it. I’m sure Cash himself, if asked this same question, would guffaw out loud at the very idea of it…and not out of false modesty, either.
Cash was a wonderful guy, and he carved out a very rewarding career. He was able to reach and entertain a great many people and probably enjoyed more genuine love than Sinatra, who I think tended to inspire awe and envy that was often confused with love, but in terms of pure musical talent and ability, Sinatra was in a league all his own.
How’s that, Ritzy? 
Sinatra by a long way. If you told me that I could never again hear him sing, say The lady Is a Tramp or Ebb Tide or I’ve Got You Under My Skin or lots of others, I would be very unhappy. Much as I like Cash’s music I would feel no loss to never hear any of it again.
The barroom brawl would never happen after Cash found the horse’s head in his bed.
Johnny all the way. Sinatra doesn’t really do it for me.
Plus Sinatra never covered a Depeche Mode song.
Cash would beat the pomade out of Sinatra in any contest you’d care to name.
In a related story, oranges are better than apples.
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Johnny Cash. Cash is The Man. 'nuff said.
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Cash again.
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Oranges.
Frank who?
Ya know, lissener, if you’re gonna criticize the wack-ass nature of the OP (which I’ll cop to), you really shouldn’t follow that up with a knock-it-out-of-the-ballpark response. It makes me dizzy.
I see what you’re saying, and in many ways you get at the crux of why I think they make an interesting apple/orange choice. And one of the reasons that I give the edge – a slight edge, in my case – to Sinatra is that when push comes to shove, I’m a fan of technical polish. Not that Cash didn’t have any polish, but he maintained a rough-hewn edge that Sinatra eliminated early in his career, if ever had that edge at all. Sinatra, in a way, is the embodiment of polish (as is the genre he worked in).
In other words, Sinatra is MOMA, Cash the American Folk Art Museum, and if I can only visit one while I’m in New York, it’ll be MOMA every time.
That said, dropping either one from my music collection would be a big loss.
You know, lately I have been enjoying the Michael Buble CD. I think he has an exceptional talent, tearing it up on many old standards. Summer Wind and Come Fly With Me are the only tracks in the collection that sound weak to me. I’m accustomed to the Sinatra readings of these tunes. As good as this Buble kid is, at this point in his career he’s not fit to hold Sinatra’s ashtray. Sinatra sang at a level of mastery so high that it sounds deceptively easy on the records - until I hear someone else do the same songs in the same style.
It would be tragic but, for this particular hypothetical Sophie’s choice, I’d have to let my Johnny Cash records go. I’m a bass/baritone, so I’d just have to grudgingly sing the songs myself.
Sinatra music is a staple for me. I learn my own story by listening to him sing about his. Most recently, I got a real kick last New Year’s Eve watching some of his TV appearances. During his performance of *One For The Road, * he would toke on his cigarette and sip his highball between phrases. Just as cool as the other side of the pillow, baby.
In a fight? Sinatra was one skinny Guinea. Cash would mess him up.