Not much—the theory of individual irreducible representations is a 'wild' problem, in some technical sense (that I don't know). My understanding, which comes entirely from informal conversations with Nat Thiem, is that the state of the art is to lump together representations until you get more nicely behaved objects called supercharacters. Nat's publications and their references should give some some idea of what's known here.

Not much—the theory of individual irreducible representations is a 'wild' problem, in some technical sense (that I don't know). My understanding, which comes entirely from informal conversations with Nat Thiem, is that the state of the art is to lump together representations until you get more nicely behaved objects called supercharacters. As far as I know, the original definitions are due to André (who calls them ‘basic characters’) and Yan, and there is an explicit supercharacter table.