Here is the full list of all 32 Just Sing trophies - there are 18 bronze, 8 silver, 5 gold and 1 platinum.
Obtain all other Trophies
Play Battle mode 30 times
Sing 100 songs in Party Mode
Get all bonuses on Star Road
Sing 45 different songs
Accumulate 500 Stars in the Star Road
Sing 50 songs in Party Mode
Play Battle mode 15 times
Upload 20 performances
Play Battle mode 5 times
Accumulate 250 Stars in the Star Road
Get 5 stars on 40 different songs with one performer
Sing 30 songs in Party Mode
Sing 30 different songs
Get 5 stars on 20 different songs with one performer
Accumulate 100 Stars in the Star Road
Upload 10 performances
Sing 15 different songs
Upload 5 performances using 5 different themes directly from the Results menu
Upload 5 performances
Sing a song with vocal track volume set to 0
Get 5 stars on 10 different songs with one performer
Upload your first performance
Sing a song in Party mode with 2 or more performers
Play Battle mode for the first time
Get 5 stars on 5 different songs with one performer
Accumulate 20 Stars in the Star Road
Get 5 stars on a song with one performer for the first time
Get 20 Perfect grades in a song
Get 10 consecutive Good or Perfect grades in a song
Select a theme and sing a song in Party mode
Sing a song in Party mode