Classic mode features branching paths, several types of challenges, and a Kid Icarus: Uprising-style difficulty system.
http://www.nintendoworldreport.com/news/38274/new-super-smash-bros-for-3ds-solo-mode-details-revealed
The Super Smash Bros. official site has been updated with several new tidbits of info regarding Solo mode play in the 3DS version of the game.
The new entry's take on the series' traditional Classic mode will feature a sort of overworld map with branching paths. Players will stop at various points on their chosen paths to encounter a host of challenge types, such as one-on-one fights, team battles with CPU players, matches where players must defeat a series of opponents one after another or in groups, and more.
Winning Classic mode battles will yield rewards, from trophies to gold to new character customization parts. A sliding difficulty scale that appears to be borrowed from Sakurai's last game, Kid Icarus: Uprising, will determine how glorious the loot will be. Players will bet gold to tip the difficulty scale towards higher intensity levels and greater rewards.
The Stadium sub-mode also returns for this game, and so far it seems that Multi-Man Smash, Home-Run Contest, and the recently-unveiled Target Blast are the only options available. The one interesting thing to note is that the Stadium mode select screen seems to suggest that Multi-Man Smash may feature Mii Fighters instead of the faceless wireframe or polygonal combatants from previous games in the series, which could inject more personality into the experience.
It is currently unknown how much overlap there will be between the feature sets of the various modes in the handheld and console versions of the new Smash Bros. For now, many of the Solo mode details discussed here have only been specified for the 3DS version.