After years spent as a globe-trotting, musician—including stints at an Israeli bar and a French ski resort—Swedish guitarist Anders Osborne finally settled down in New Orleans two years ago. And New Orleans is finally taking notice.

In April, Anders and his Osborne Orchestra won two major music competitions in town: the Abita Beer/OffBeat Magazine Open Blues Jam at Mid City Bowling Lanes, and the WWL-1V Jazz Search.

As Jazz Search winner, Osborne earned a slot in the Jazz and Heritage F...