You can play with a number of different controllers in the Wii U version.

Wii U GamePad

If you are playing alone, you can play using only the Wii U GamePad screen.

Wii Remote Plus

You can play by holding it sideways. You can use a normal Wii Remote as well.

Wii Remote Plus + Nunchuk

Wii Remote Plus + Classic Controller Pro

The Classic Controller is also supported.

Wii U Pro Controller

Nintendo GameCube Controller

You can connect up to four GameCube Controllers using one GameCube Controller Adapter for Wii U. Connect the adapter to the Wii U console using two USB cables. You can use the USB ports on the front or the back of the Wii U.

Nintendo 3DS and 2DS systems

You can use Nintendo 3DS and 2DS systems as controllers, as long as you have a copy of Super Smash Bros. for Nintendo 3DS for each system.
If you don't have Super Smash Bros. for Nintendo 3DS, you can purchase and download the Smash Controller software, available in Nintendo eShop from June 14th 2015, for £0.89.
Please note: compatible with Smash mode only.