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