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