Here is the full list of all 41 Inertial Drift trophies - there are 22 bronze, 15 silver, 3 gold and 1 platinum.
Earn gold in a time trial
Earn gold in an endurance event
Compete in an endurance event
Win a duel
Compete in a duel
Win a race
Achieve 100% on a story mode playthrough
Complete Ada's story
Compete in a race
Complete an event on a reverse track layout
Compete in a time trial
Complete the tutorial
Use the Drift Stick
Earn all trophies
Compete in a ghost battle
Compete in a style event
Set a leaderboard time in Arcade Time Attack
Set a leaderboard score in an Arcade Style event
Unlock the Boba
Unlock the Katana
Unlock the 7D
Unlock the Bullet
Unlock the Ventus
Unlock the Jet Z 880
Unlock the Raptor
Unlock the Gecko RX
Unlock the Jester
Unlock the Gladio
Unlock the Impuls
Unlock the Supreno
Complete Edward's story
Complete Ibba's story
Complete Viv's story
Complete a story play through on "Xtra Crispy"
Complete 10 laps in practice mode in one session
Beat the target time in a ghost battle
Earn gold in a style event
Complete an event on every track
Hit a checkpoint with less than 1 second left
Earn a platinum medal
Complete grand prix mode with every car