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