Here is the full list of all 42 Knack trophies - there are 28 bronze, 9 silver, 4 gold and 1 platinum.
Received all trophies
Completed all chapters on Very Hard difficulty
Completed all chapters on Normal difficulty or higher
Completed all chapters on Hard difficulty or higher
Completed a stage without taking any damage on Hard difficulty or above
Achieved five-star ranks in all Coliseum Attack stages
Completed a Coliseum Attack stage
Achieved five-star ranks in all Time Attack stages
Completed a Time Attack stage
Defeated all bosses using Diamond Knack
Defeated Gundahar's Robot using Massive Knack
Defeated Katrina's Mech using Sunstone Knack
Defeated the Giant Orb using Dark Knack
Defeated the Chieftain's Mech using Brittle Knack
Cleared a chapter as Vampire Knack
Unlocked all upgraded Knack characters
Acquired all of the Gadgets
Unlocked an upgraded Knack
Completed a chapter only using Super Moves on enemies and without dying once
Defeated a boss without taking damage
Achieved 50 consecutive hits with the Combo Meter equipped
Defeated the three Gatekeeper Goblins and cleared Chapter 3-2 without dying
Defeated 8 or more enemies with a single Super Move
Destroyed the 3 anti-aircraft guns in Chapter 8-2 within 5 minutes
Acquired a total of over 100,000 relics
Defeated the Chapter 13-5 boss
Discovered all of the treasure chests
Defeated the Chapter 11-5 boss
Defeated the Chapter 7-5 boss
Defeated 5 gunships by throwing cars
Achieved 20 consecutive hits with the Combo Meter equipped
Constructed a Gadget
Defeated 20 enemies as Stealth Knack and completed the chapter
Landed 100 hits using the Dash Attack
Collected the clear crystals in Chapter 4-2 without being detected by the security robots
Defeated the Chapter 3-5 boss
Defeated 100 robots
Completed a non-boss chapter without using any Super Moves
Defeated 100 goblins
Acquired a Crystal Relic
Acquired a Gadget part
Discovered a treasure chest