Karts and Bikes

Bikes
Small:
Bubble Bike: Win Mirror Leaf Cup
Magikruiser: Play 8 different courses in Time Trials
Quacker: Win 150cc Star Cup

Medium
Dolphin Dasher: Win Mirror Star Cup
Nitrocycle: Attain 1 star in all 100cc Nitro Cups
Rapide: Win 100cc Lightning Cup

Large
Phantom: Win Mirror Special Cup
Torpedo: Unlock 12 Expert Staff Data in Time Trials
Twinkle Star: Win 100cc Star Cup 

Karts
Small
Blue Falcon: Win Mirror Lightning Cup
Cheep Cheep Charger: Attain 1 star in all 50cc Retro Cups
Rally Romper: Unlock 1 Expert Staff Data in Time Trials

Medium
B Dasher: Unlock 24 Expert Staff Data in Time Trials
Glory: Unlock 24 Expert Staff Data in Time Trials*
Turbo Blooper: Win 50cc Leaf Cup 

Large
Dragonetti: Win 150cc Lightning Cup
Piranha Prowler: Win 50cc Special Cup
Aero Glider: Attain 1 star in all 150cc Retro Cups

*I have no idea why the B Dasher and Glory have the same unlockable!