So there is a growing movement of modern songs done in a 20's-40's style. The most prolific of these is Post Modern Jukebox. This song isn't from them, but her voice is tremendous!!
Robyn Adele Anderson - Clint Eastwood (originally by The Gorillaz) Video
In the time it took to list that one ^, one of my most favorites came on:
Haley Reinhart - 7 Nation Army (originally by White Stripes) Video
She is as beautiful as her voice. This is a Postmodern Jukebox production.
Post Modern Jukebox again. Be careful, lads. Once you start into these videos you are going to get hooked. They are very addictive.
Casey Abrams, Haley Reinhart, Morgan James, Ariana Savalas - All About That Bass (originally by Meghan Trainor) Video
PMJ:
Casey Abrams (who sounds eerily like Jack Sheldon of School House Rock fame) and the "Sole Sisters" tap dancing - Umbrella (originally by Rihanna) Video