Boss Attack - Complete Arcade mode once
Caravan - Complete Arcade mode once
To unlock these stages, simply clear each stage with an A rank or better for Stage Omega, or a C rank or worse for Stage Alpha. The best way to get a good ranking is to get all quickens as fast as possible, then make huge combos. The best way to get a bad ranking is to not do anything at all. Simply dodge the enemies and bullets, then defeat the boss. It's still alright to get quickens, but only quickens, no combos.