My answers? I'm going to be strongly swayed towards an older S&W for virtually all DA purposes.
I think the Python is a handsome sixgun but I've never seen a 586 I didn't like. Or a model 19 for that matter. Add to that model 27's and .38/44's.
Same for daydreaming.
For a dedicated hunting sixgun a later model S&W will be more adept because they're drilled and tapped for scope mounts. Any of them will do for iron sight hunting but again, my preference would be S&W.
For accuracy, the Python wins hands down.
For action shooting, the S&W would be first pick.
For throwing in the truck or tackle box to be forgotten about, a stainless Ruger wins easily.
For concealed carry, any would do with a shorter barrel. S&W's and Rugers are easier to come by in short lengths. Either S&W or Ruger will do fine.
For duty use, probably S&W.
S&W is the only one of the three that readily accepts optics.
Ruger probably has an edge in durability but I think a lot of the rhetoric about Rugers being "built like a tank" and indestructible are overblown.
The Ruger is really the best platform for custom work. It is the only one that can be converted to a five-shot big bore .41Mag or .44Spl. While the L-frame can be rechambered to .41Spl or 10mm.
This is one instance where Colt's resale value is definitely ahead. Unlike SAA's, Pythons are no longer made and only going up. Of course, this assumes that you don't pay a collector premium today and expect to make a sizable profit in five years.
The Ruger is probably the best value. They're still quite affordable. New S&W's are way too high for what is no longer a better gun. Older S&W's are only going up. Good luck finding a good deal on a Python.
All that said, the 27 is probably my favorite double action .357. Followed closely by the 586, which for me would need new sights. I also love the model 19 but would load it more moderately. I have to say that a blued GP cut back to 5" (hate Ruger's angled underlug), slicked up action, fitted with custom sights, a bright blued finish and fitted with Herrett's Roper stocks would be strongly in the running but that's a lot to invest to get the same thing you get with a box stock 586.
My favorite .357 of all would be the Ruger Old Model flat-top Blackhawk in the 4 5/8" length.