To be honest I'm not a big fan of Eeveelutions because of the heavy hype and promotions they get. Still, it's the same reason why I dislike Charizard with a passion.
For looks, I go for Sylveon. It looks too... I don't know how to phrase it, but feminine. Well the same could be said for its leafy and icy counterparts, but Sylveon is much more extreme. I have no problem with feminine Pokemon, before anyone gets me wrong, but I think Eeveelutions should be able to look both male and female.
For competitive viability, Flareon followed by its Sinnoh counterparts. Leafeon faces competition from Sawsbuck, who gets STAB Return and Horn Leech, while its defensive variants can face competition from Exeggcutor. Glaceon has a stiff SR weakness, and perma hail being banned isn't doing it any favours. Flareon... lol it's pretty famous for it's movepool.