Here is the full list of all 15 Drive!Drive!Drive! trophies - there are 13 bronze, 1 silver and 1 gold.
You completed Campaign mode!
You jumped more than 300m.
You smashed into an opponent from another track.
You drifted more than 200m.
You drove around a corkscrew.
You came 1st in all tracks in a race.
You drove around a loop the loop.
You jumped more than 100m.
You obtained a medal of the golden variety.
You smashed into an opponent from another track in mid-air.
You smashed into an opponent in mid-air.
Yalopolooza! You smashed an opponent up.
You gave an opponent a piggy back.
You completed Offensive mode! That's pretty hardcore.
You drifted more than 400m.