To listen to alternative music tracks on select stages press and hold either one of the two analog trigger buttons (L or R) when picking your stage. The following stages have alternate music tracks available:

Great Bay: Saria's Song
Hyrule Temple: Fire Emblem
Icicle Mountain: Balloon Fight
Big Blue: Mach Rider
Yoshi's Island: Super Mario Bros. 3
Onett: Earthbound (2)
Pokemon Stadium: Pokemon Battle Theme
Mushroom Kingdoms I and II: Dr. Mario
Battlefield: Multi-Man Melee Theme
Final Destination: Alternate SSBM Theme