It tells you on the eShop which controllers work for which games. When you select a game and view it's profile, scroll down to the "Compatible Features/Accessories" section and tap View. It tells you what controllers you can use.
In Kirby Super Star's case:
* Wii U Game Pad * Wii U Pro Controller * Wii Remote + Classic Controller Pro
Note that Classic Controller Pro also refers to the standard Classic Controller, since they have the same buttons, just in a different placement.
All Wii U VC games can be used with the same controllers, though SNES VC games require you to hold something (I forget what) to boot them up with just a Wii Remote, and you may run into possible issues due to not having enough buttons with some games.