Here is the full list of all 12 VR Karts trophies - there are 8 bronze, 3 silver and 1 gold.
Win an online race
Compete in an online race
Win every race in Rookie, Amateur and Pro Cup championships
Win the Pro cup
Win the Amateur cup
Play with alternate controls
Win a race in Turbo mode
Record a lap time in Time-Trial
Customize character
Win the Rookie cup
Win a race
Complete tutorial