Here is the full list of all 40 Accel World vs. Sword Art Online trophies - there are 26 bronze, 10 silver, 3 gold and 1 platinum.
Obtained all trophies.
Raised a weapon from Rank 1 to Rank 10.
Acquired 100 total titles.
Raised one AW character's skill level to 500.
Enhanced one weapon to +10.
Completed all of Hexe's requests.
Raised one SAO character's skill level to 500.
Unlocked all seven Kings.
Received 50 titles.
Raised one character's level to 1000.
Raised one character's level to 500.
Destroyed a solid Otherworld object.
Cleared 50 quests.
Unlocked Nitride Unica.
Started episode 5 of "Say It, Miss Strea!"
Cleared "Timeless Bonds."
Cleared every "Mystification" episode.
Cleared every "Navigation" episode.
Unlocked Seven.
Cleared "Millennium Twilight."
Cleared "A Gleam of Hope."
Chained together 8 skills with Skill Connect.
Obtained 100 different types of equipment.
Cleared an Extra Quest with only Sky Raker, Ardor Maiden, and Aqua Current.
Cleared an Extra Quest with only Asuna and Black Lotus.
Cleared an Extra Quest with only Kirito and Silver Crow.
Cleared "Lost Wish, Tumultuous Heart."
Cleared "Showdown."
Cleared an Extra Quest.
Cleared "Disastrous Moonlight."
Cleared "Cold Trap."
Used BP to buy something.
Cleared "The Impatient Skies."
Cleared a Friend Quest.
Cleared a Challenge Quest.
Cleared a Sudden Quest.
Employed a blacksmith's services.
Cleared "The Invader."
Defeated the aura-clad monster general.
Performed a Switch.