First band I thought of was The Kills: "No Wow", "I Hate The Way You Love" and "Rodeo Town" are all pretty great.
I was thinking Billy Bragg too. And I'd mix in Bahamas, although sometimes drummer and backup singers with him.
St. Vincent Tiny Desk is one of my favorites Also, maybe this Kingfish Tiny Desk? He brings a bass player along though so it might disqualify.