Here is the full list of all 14 Shisen-Sho NIKAKUdori trophies - there are 13 bronze and 1 gold.
Earn 50 Crown in challenge mode
Earn 100 Golden Crown in challenge mode
Complete All Stages with SETTING1 in Free mode
Complete All Stages with SETTING2 in Free mode
Get a 45 Combo in Free mode
Get a 68 Combo in Free mode
Complete 10 Seconds in Free mode
Complete 5 Seconds in Free mode
Complete 1 Seconds in Free mode
Complete 0 Seconds in Free mode
Complete with 3000000 Points in Free mode
Complete with 4500000 Points in Free mode
Complete with 5500000 Points in Free mode
Complete Record mode