Here is the full list of all 50 Naruto: Ultimate Ninja Storm trophies - there are 37 bronze, 11 silver, 1 gold and 1 platinum.
Prevented an opponent's Extra Hit.
Completed the Genin Training Arc.
Finished an opponent with an item.
Created a new Ichiraku Ramen.
Finished an opponent with a Support Combo.
Finished an opponent with a throw.
Performed six or more successful rope actions.
Finished an opponent with Support Ninjutsu.
Attacked an opponent with a Supportive Extra Hit.
Finished an opponent while Awakened.
Finished an opponent with Ninjutsu.
Interrupted an opponent's Ultimate Jutsu.
Finished an opponent with an Ultimate Jutsu with the Storm Gauge at MAX.
Finished an opponent with an Ultimate Jutsu.
Acquired all the trophies.
Completed the Chunin Exam Arc.
Completed the Leaf Destruction Arc.
Completed the Tsunade Search Arc.
Completed the Chasing Sasuke Arc.
Completed over 10% of the missions.
Completed over 20% of the missions.
Completed over 30% of the missions.
Completed over 40% of the missions.
Completed over 50% of the missions.
A new item was added to the shop.
Six or more different items were added to the shop.
Acquired a Support Ninjutsu Book.
Acquired eight or more Support Ninjutsu Books.
Acquired a Ninja Handbook.
Acquired 14 or more Ninja Handbooks.
Created 10 or more different Ichiraku Ramens.
Acquired a Figure.
Acquired over 30% of all Figures.
Acquired over 50% of all Figures.
Acquired an Ultimate Jutsu Movie.
Acquired over 30% of all Ultimate Jutsu Movies.
Acquired over 50% of all Ultimate Jutsu Movies.
Acquired a music file.
Acquired over 30% of all music files.
Acquired over 50% of all music files.
Destroyed a door with a Rasengan.
Destroyed five or more objects with one shuriken toss.
Destroyed 10 or more objects with one shuriken toss.
Cleared 12 or more rings with one Naruto Cannon.
Cleared 24 or more rings with one Naruto Cannon.
Cleared 42 or more rings with one Naruto Cannon.
Finished an opponent with an Ultimate Impact.
Finished an opponent with an additional strike after downing them.
Finished off an opponent from a wall.
Won a Jutsu Clash.