Too late to help this year, but I've found that the Pioneer Square scene has the most crowds/craziness. You also get more frat boys than I enjoy partying with, but still. The New Orleans restaurant (114 First Avenue South) always has a great band and some decent food/drink specials. It's been a decade or so since the Pioneer Square crowds got so out of hand people were severely beaten and at least one person died. It was super bad news. The ghosts of that evening still hang around, and police keep crowds from partying in the street now.
If you're just looking for a New Orleans music club ATMOSPHERE, all year round, the Highway 99 Blues Club is fantastic. http://highway99blues.com/ The vibe here is lovely. "Red House" (a local band of artists who have been doing this for years and have the chops to prove it) is always a solid choice. They've even got a Zydeco band coming next week. There are many tables and chairs, a nice dance floor to the side of the stage.
For Mardi Gras celebrations, in general, almost every bar/dance club will try to capitalize on this event (same goes for St Patrick's Day, Cinco de Mayo, etc). I imagine you'd find beads and masks, and maybe Hurricane drink specials, at most places.