I would highly recommend just strolling up Frenchman St, in the French Quarter, and poking in your head into whatever place seems to be happening, or fits your taste. There are SO many options here, and the whole street is bustling with an energetic music scene EVERY night of the week (I kid you not). I’d advise seeing Jazz at The Spotted Cat or Apple Barrel for an authentic, crowded, and energetic musical experience. It was so great too see young people in their twenties playing there asses off and doing traditional music. No-cover (1 drink minimum).

For a little nicer atmosphere, you can also try the Three Muses across the street on Frenchman, which has great live music in a beautiful but small (everywhere gets REALLY crowded, so arrive early) room that actually serves it’s drinks in a glass (as opposed to a plastic cup) along with food. Snug Harbor has a cover, but gets some higher profile Jazz artists in a more gentile atmosphere. Also of note is the Maison for its more dance oriented music programming (swing, funk, etc.).

I also saw an excellent show at the re-purposed U.S. Mint in the Quarter that my son and I stumbled upon when going to checkout an exhibit on Preservation Hall. They have a FREE concert series that they put on in a beautifully designed, acoustically treated hall with a state of the art sound system and video projection (in a Mint of all places!). It is part of a special commissioning program sponsored by the National Parks system (gotta love a place that has a National Park of Jazz! I love that my taxes go to this. ). Here’s the calendar:
http://www.nps.gov/jazz/upload/March-2012-3.pdf