Here is the full list of all 29 Destruction AllStars trophies - there are 6 bronze, 17 silver, 5 gold and 1 platinum.
Complete and Win the Carnado Tutorial
Complete and Win the Mayhem Tutorial
Complete the vehicle emote tutorial in Open Training with any character
Unlock all Destruction AllStars Trophies
Earn all of the star objectives for a single event
Earn all star objectives in Ultimo Barricados' Series, Mutual Respect
Earn an S Rank Wreckognition rating or higher in any event
Complete and Win the Gridfall Tutorial
Complete and Win the Stockpile Tutorial
Perform 100 Wrecks in any mode
Wreck an opponent's hero vehicle in an online match
Deal 100 damage whilst stealth in Cypher, without taking any damage in an online match
Block 100 damage with the Undisputed's shield in a single use of the breaker in an online match
Keep a full shield in Gravitron for 5 seconds in an online match
Slice 2 vehicles in a single slice with Sabre in an online match
Set an entire team on fire with Cerberus in an online match
Set 3 AllStars on fire at once with Wildfire in an online match
Wreck 3 Rivals using CRASHendo in an online match
Wreck 3 opponents with a single use of Mr. Sparkles' breaker in an online match
Take out 3 targets in a single use of Xero's X-Ray in an Online Match
Take out 3 targets in a single use of Number One's Lock on in an online match
Get a Team kill with Barong's Countdown in an Online match
Shred 3 cars in a single use of the Shredder in an online match
Spike 3 Rivals at once with the Morningstar in an online match
Attach drones to 3 Rivals with The Boxmobile in an online match
Smoke 3 Rivals with the Smoke Commander in an online match
Reach 135 MPH with Callisto in an online match
Win 50 online matches
Win at least 1 online match with each AllStar