Vaporeon: I love its stock Sugimori art and prefer most of its past sprites. I used to consider it my favorite, but I don't use it in the 3DS games because its model looks weird (the neck area seems too short).
Jolteon: This one's always been one of my least favorite Eeveelutions. Not only does it not look cuddly at all, but it's also one of the most dog-like members of the family and I've never considered myself a dog person. I do like some of its art, though.
Flareon: The most adorable in the anime and I really like a lot of its TCG art (Jungle Flareon is one of my favorite card illustrations). I'm not much for using it in-game though, despite evolving my only full-odds Shiny Eevee into one.
Espeon: I do enjoy its elegant cat-like design and found myself trying to collect its cards because it reminds me of Mew (the main cards I try to collect), but I haven't really used it much in-game.
Umbreon: I used to adore Umbreon, but I find myself liking it a lot less these days. Although its stock Sugimori art looks really cool, none of its sprites have ever been that attractive IMO but what ultimately killed it for me was its appearance in the Eevee and Friends special that aired a few years ago.
Glaceon: There was a time that I absolutely despised Glaceon. Terrible stock art and initial sprite (I still hate that they based its BW sprite on the DP sprite instead of the Platinum one) and I always thought its color scheme seemed too dark. However, I fancied myself an Ice-Type trainer when I MM'd a Shiny Eevee in B2 and decided to evolve it into Glaceon so I could try getting used to it. Now I adore Glaceon, but only its Shiny form (anime is also fine since it's almost identical to its Shiny palette). It's also my favorite Eeveelution to use in battle.
Leafeon: I really enjoy it design-wise and would probably lean more toward it IRL, but that's about as far as it goes. I wish they'd done something more interesting with its Shiny...
Sylveon: Probably my least-favorite of the batch. I've nothing against cute pink things (several of my favorites are pink), but there's something kind of eerie about it and the fact that it's the largest Eeveelution bothers me for some reason. Also too dog-like for me (perhaps the most dog-like of the group).