This shirt is denim* blue and does up with white snaps. That’s really all we know about it. I can’t really tell if it’s actually denim or just that colour blue, but odds are it’s probably denim. It has breast pockets. (*shirt isn't actually denim, just a similar denim-blue - see first comment.)
Brand: Levi's (blue washed chambray shirt)
In Baby (11x04), Dean wears this shirt when they leave for Oregon and keeps it on until the end of the episode, except when he very briefly wears a suit when leaving the police station.
In Beyond the Mat (11x15), Dean wears this shirt at the beginning of the episode while in the Bunker.
In All in the Family (11x21), Dean wears this shirt the morning after they arrive back in the Bunker with Chuck, and he and Sam have a conversation about what it’s like having God in the house.
In Ladies Drink Free (12x16), Dean wears this shirt at the beginning of the episode when he and Sam are in the BMoL headquarters, and as they drive to the hunts location, until they check into the motel.
In All Along the Watchtower (12x23), Dean wears this shirt for the entire episode.
FATE: In Let the Good Times Roll (13x23), Michael changes out of this shirt, and given Michael doesn’t share Dean’s fashion sense, very likely would have just ditched it completely.