Go with the American Eagle.
All are probably fine, but in my experience Federal American Eagle is the best-quality "blasting ammo" brand from a major maker. Their FMJ offerings in 9mm, .380, and .38 Special are all recommended. Clean, uniform, accurate, nice brass -- there's just nothing wrong with it.
I also like their flat-point .357 Magnum JSP a lot. It is accurate plinking ammo for lever carbines, and there's no reason you couldn't hunt with it.
Back to your question. I've shot cases of Fed AE in 9mm (124 grain), and have also tried Winchester and Blazer Brass in that caliber.
I would rank as follows:
1) Federal AE
2) Blazer Brass (quite good, close to AE)
3) Winchester
Not much experience w/ Remington UMC in 9mm. In other calibers I tend to find it a notch below the other brands listed.
PS: For premium ammo -- carry or hunting -- I generally strongly prefer CCI/Speer, Hornady, Winchester, or even Remington (Golden Sabers and the Express lead .38+P hollowpoints). Not a big fan of Federal. But at the "non-crappy blasting ammo" price point, Federal (and particularly the AE line) has been very good.