Here is the full list of all 37 NOW That's What I Call Sing achievements worth 1,000 gamerscore.
You have finished your first song
You have started By Heart mode
You got a score between 50,000 and 51,000 points
You have started Pass the mic mode
Sing all the songs
Start Duet mode
Start Expert mode
Start 20,000! mode
Start Elimination mode
Accumulate 50 VoxPoints
Accumulate 100 VoxPoints
Accumulate 500 VoxPoints
Get a score over 80,000 points
Get a score over 100,000 points
Get a score over 120,000 points
Get 500 'PERFECT'
Finish a song in Elimination mode
Finish a song in 20,000! mode in under 60 seconds
Finish with a similar score to your partner in Duet mode
Get 10 'PERFECT' in a row in Expert mode
Get 'PERFECT' for the parts of a song with no lyrics in By Heart mode
Finish 15 songs in a session
Listen to 10 songs in a row in Jukebox mode
Finish 40 songs
Finish 90 songs
Finish 150 songs
Sing 3 songs in a row from before 2008
Sing 5 songs in a row released after 2013
View all the credits
Unlock all the game modes for a song
Unlock all the game modes for all songs
Browse through the song covers 3 times in less than 20 seconds
Get a score inferior to one point
Finish 5 songs in Expert mode without the vocals
Pass the mic 30 times in Pass the mic mode
Finish a song in Expert mode without the vocals
Hit all the golden notes in a song