Diamond Cup:
Get first place in the Ruby, Sapphire and Emerald Cups on Standard or Expert difficulty.

Master Class:
Get first place in the Ruby, Sapphire and Emerald Cups on Expert difficulty.

Bahumut vehicle:
Take an F-Zero GX License Card to an F-Zero AX arcade machine. Save the data that is unlocked to a memory card.

Dark Schneider vehicle:
Beat Story mode.

F-Zero AX machine parts:
Beat Story mode on Hard difficulty.

F-Zero AX racers:
Beat Story mode on Very Hard difficulty. You can also finish an F-Zero AX arcade machine race and save the data to a memory card. Now load that save and you can purchase the racer at the shop.

F-Zero AX tracks:
Beat Grand Prix mode on Master difficulty.

Snaking:
It's arguable whether or not snaking is a glitch, but when performed correctly, it gives you a huge speed boost. To perform the trick, alternate between hitting L and R while tilting the control stick in the same direction. Your machine will slide from left to right and accelerate beyond its normal top speed. This trick works best when using heavy machines with low grip ratings (ie: Black Bull). Also, it helps to set your machine to maximum acceleration. Note that each left to right motion must be done in quick succession both to minimize drift and to obtain the highest possible speed.

Thanks to Revolution readers Bugrle Bargle, Johann Sven, longday, dogdog11, Jimmy Dean and Rupture!