Nope, Ash hasn't caught an Eevee so far, and it isn't one of the Pokemon that he has trained or worked with either (as of May 2014).
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No, Ash will not catch Meloetta.
Actually you cant catch eevee Bill gives him to you at Goldenrod city
There is no such thing as a shiny eevee in the trophy garden.
I'm not sure if you can catch Eevee in Pokemon pearl but you get it as a gift from Bebe in Hearthrome City after you get the national dex.
You can't catch an evee anywhere, but you cn get one from a lady in Hearthome and you can catch one in the Mansion's garden but nowhere else. Sorry.on route 14 you can find plasma eevee and curse eevee and all other sorts of eevees even her evolvesions