Beat the last Bowser
Find every normal and secret flag (there are 94 in all)
Catch Nabbit when he appears in each of the first seven worlds
Obtain every star coin in the standard 8 worlds
Obtain all star coins in every world and level, including level 9 in Superstar Road
New Super Mario Bros U physics
Beat the final boss.
Play as Nabbit in Single Player
To play as Nabbit on any stage in single player, hold down ZL when selecting a stage, and Nabbit will replace Luigi for that stage. Luigi will return after you finish the stage, or Nabbit dies in the stage.
Play as your Mii
If you use the secret single player Nabbit code on the final Star Road level you will play as your Mii instead of Nabbit.