2nd analog stick or BUST. This means Wiimote + Nunchuck or just Wiimote = no go.
Wii U Pro
GameCube
Wii U GamePad
Classic Controller
Any of these are fine. The best option of them? GameCube, strictly because of the wire.
2nd stick is NECESSARY for high level play. B-sticking, continual momentum reverse aerials, etc.
The trigger buttons were important in Melee because Melee had light shields vs. heavy shields, and the triggers being springed would trigger the difference if you lightly pressed on it versus "clicking" it in. Brawl and SSB4 don't have light shields, only heavy shields, which are activated either way, so that no longer matters.
Other than that it's just preference of ergonomics.
For me, I love how the Y, X, and B buttons wrap around the A button the GCN. It makes for much, much easier execution on certain actions. The diamond setup for the buttons on other controller is rather limiting in my opinion.