Here is the full list of all 38 Riff Racer - there are Race Your Music achievements.
Drive 5 cars
Drive 50 songs
Unlock 5 packs
Beat a friend's ghost
Author 5 tracks
Reach Driver Level 5
Drive 25 songs
Jump more than 8 beats in one manoeuvre
Play a Pop Song
Enter the super-sync zone 10 times in under 40 seconds
Play a Classical song
Author 1 track
Drive 10 songs
Boost 20 times or more in one song
Beat a recommended ghost
Play a Rock/Metal song
Drive a whole song without hitting a barricade
Play a Hip-Hop/Urban song
Stay 100% in sync for a whole song
Play a Dance song
Roll more than 4 beats in one manoeuvre
Drive 10 cars
Drive ALL cars
Drive 100 songs
Author 10 tracks
Reach Driver Level 10
Reach Driver Level 15
Reach Driver Level 20
Stay in the super-sync zone for 25 consecutive seconds
Play a single song 10 times
Unlock 10 packs
Unlock 20 packs
Unlock ALL packs
Score more than 100,000 on a song under 3 minutes in length
Race on a song longer than 10 minutes in length
Smash 30 boxes or more without missing one
Drift more than 16 beats in one manoeuvre
Play a Jazz/Funk/Soul song