There are 42 Trophies that can be collected.

Bronze

That Will Suffice
Finished the [Tutorial] mode.

Time to Hit the Arcades
Finished [Arcade] mode.

Mars Need Winnin’
Attempted [Unlimited Mars] mode over 10 times.

Kaka-kaze
Have been blown up by the chibi-kakas in [Unlimited Mars] mode.

Eye of the Tiger
Trained for more than 30 minutes, non-stop.

Mastered the Basics
Cleared over 50 missions in [Challenge] mode.

Just the Beginning
Defeated the boss level at level 100 of the [Abyss].

About Half Way
Defeated the boss at level 500 of the [Abyss].

*Wheeze*…Just Getting Started
Increased all character stats to 50 in [Abyss] mode.

Oh—NOW I Get It…!
Finished the Calamity Trigger scenario in [Story] mode.

Breaking in the New Guys
Finished Makoto’s, Valkenhayn’s, Platinum and Relius’s scenarios in [Story] mode.

Mind on my Money
Accumulated over 100,000P$.

Artwork?! Gimmie!
Unlocked 50% of the items in the [Gallery].

I See What You Did There
Played back replay data over 10 times in [Replay Theater].

There’s No “I” In Team
Won a team elimination match without even fighting.

Level 20
Reached level 20.

I can’t Stop Winning
Won three [Ranked Matches].

Like a Boss
Fought 100 matches in [Ranked Match].

Plays Well With Others
Fought ten team [Player Matches].

Feels Like the First Time
Won your first team [Player Match].

Silver

Murakumo…AWAKEN!
Defeated the secret boss in [Arcade] mode.

The Beginning of the End
Finished [Score Attack] mode.

Mastered the…Master?
Cleared over 100 missions in [Challenge] mode.

Getting Down to Abyss-ness
Defeated the boss at level 999 of the [Abyss].

To Be Continued—!
Witnessed the true ending in [Story] mode.

Just Getting Used To It
Raised any character’s PSR to over 200 in a [Ranked Match]

Gold

Abyssmal Shopaholic
Acquired all the items from the shop in [Abyss] mode.

Gallery Guru
Unlocked everything there is to see in the [Gallery].

Platinum

Un…believable
Mastered BlazBlue Continuum Shift – Extend.

13 Secret

http://www.ps3trophies.com/forums/playstation-trophy-news/10311-blazblue-continuum-shift-extend-trophy-list.html