I've caught all three of them at one point, but I don't believe I've ever used them much in-game, aside from perhaps Azelf and Mesprit.
In terms of favorites, competitively, I would say I like Azelf better. It's got amazing Speed, if I'm not mistaken, and it also has access to some moves which we might find to be rather helpful, such as Stealth Rock, U-Turn, and so forth. Other two members of the trio, in comparison, aren't that great, as unfortunate as it may sound.
Design-wise, my favorite would likely have to be Uxie. Its color palettes consist of light blue, grey, and yellow, which, at least in my eyes, seem to complement each other quite beautifully. If it weren't for that, I probably wouldn't have felt the same about for this Pokémon. Second is definitely Mesprit.