There are 32 Trophies that can be collected.

Bronze

Monster Summon
Summon a monster in Singleplayer Mode.

First Victory
Clear Tutorial Stage 1 in Singleplayer Mode.

Beginner
Clear all Tutorial Stages in Singleplayer Mode.

Fire Jewel
Obtain Fire Jewel in Singleplayer Mode.

Water Jewel
Obtain Water Jewel in Singleplayer Mode.

Wood Jewel
Obtain Wood Jewel in Singleplayer Mode.

Light Jewel
Obtain Light Jewel in Singleplayer Mode.

Dark Jewel
Obtain Dark Jewel in Singleplayer Mode.

Offense Formation
Acquire Offense Formation in Singleplayer Mode.

Defense Formation
Acquire Defense Formation in Singleplayer Mode.

Wing Formation
Acquire Wing Formation in Singleplayer Mode.

Separate Formation
Acquire Separate Formation in Singleplayer Mode.

Serpentis
Visit Serpentis in Singleplayer Mode.

Hydrae
Visit Hydrae in Singleplayer Mode.

Leonis
Visit Leonis in Singleplayer Mode.

Scorpie
Visit Scorpie in Singleplayer Mode.

Wtich's Abode
Visit Witch's Abode in Singleplayer Mode.

Fusion Novice
Conduct Card Fusion once.

World
Conduct battle once in Multiplayer Mode.

Silver

Card Collector
Exceed 100 cards.

Dragon Master
Acquire Expansion Pack, Legendary Dragon.

Legendary Card
Acquire one card of Legendary rarity.

Practitioner
Win 100 battles.

Scholar
Obtain 50 different titles.

Legendary Master
Defeat the Demon King in Singleplayer Mode.

Gold

Warrior
Become Class 100.

Veteran
Win 100 consecutive battles.

Professional
Win 100 battles in the Ranking Match.

Title Collector
Obtain 100 different titles.

Avatar Collector
Obtain 30 different avatars.

Elemental Master
Clear all stages in Singleplayer Mode.

Platinum

All Complete
Obtain all Elemental Monster trophies.

http://www.ps3trophies.org/game/elemental-monster-online-card-game/trophies/