Earn all the prize bubbles on the indicated level to get a 100% completion and unlock the corresponding bonuses:

Anteater Concept and Anteater Concept With Frame: The Mines
Bad Witch Concept, Bad Witch Concept With Frame: The Bunker
Big Cat and Big Cat Concept With Frame: Burning Forest
Big Sumo, The Islands Concept, and The Islands Concept With Frame: Sensei's Lost Castle
Boom Town Concept and Boom Town Concept With Frame: Boom Town
Boss Concept, Boss Concept With Frame, and a Monster Truck: The Construction Site
Bunny Tail and The Gardens: Skate To Victory
Circus Concept, Circus Concept With Frame: The Frozen Tundra
Costumes Concept and Costumes Concept With Frame: Elephant Temple
Early Sackboy and Early Sackboy With Frame: Lowrider
Fairy Tale Concept and Fairy Tale Concept With Frame.: Get a Grip
Grabbing Machine, Rock God, and Rock God With Frame: Subway
Graveyard Concept, The Wedding Concept With Frame, and Skulldozer: Skulldozer
Jumping Tank, Very First LittleBigPlanet concept, Very First LittleBigPlanet Concept With Frame: The Collectors Lair
LittleBigPlanet Concept With Frame, Background Environment Concept: The Dancers Court
LittleBigPlanet Cosmos Concept, Magicians Box, and Cosmos Concept In A Frame: Great Magicians Palace
Savannah Concept and Savannah Concept With Frame: Swinging Safari
Sheriff Zapata's Explosives Machine, The Mines Concept With Frame, and The Mines Concept: Serpent Shrine
Temple Concept, Temple Concept With Frame, and a Mystical Dragon: Endurance Dojo
Terrible Oni, Islands City Concept, and Islands City Concept With Frame: The Terrible Volcano
The Collector Boss, The Collectors Pod, and The Collector: The Collector
The Gardens Concept and The Gardens Concept With Frame.: First Steps
The Savannah and The Pink Scarf: The Meerkat Kingdom
Theme Characters Concept With Frame and Theme Characters Concept: The Darkness
Wrestling Ring With Frame and Wrestling Concept: The Wedding Reception

Successfully complete the indicated level without dying to unlock the corresponding costume:

Bunny Tail: The Collectors Lair
Chicken Beak: The Dancers Court
Chicken Gloves: Great Magician's Palace
Chicken Tail: Elephant Temple
Cowboy Boots and Leather Gloves: Boom Town
Cowboy Hat: Serpent Shrine
Dinosaur Mask: The Construction Site
Dinosaur Tail: Lowrider
Googly Eye Glasses: Sensei's Lost Castle
Green Sock Puppet: Subway
Japanese Festival Headband: The Terrible Oni's Volcano
Japanese Festival Robe: Endurance Dojo
Jeans with a Belt and Cowboy Bandana: The Mines
Moustache: Burning Forest
Neon Helmet: The Bunker
Pirate Hat: Skate To victory
Pirate Hook and Pirate Eye Patch: First Steps
Pirate Waistcoat and Pirate Shorts: Get A Grip
Ringmaster Jacket: Swinging Safari
Ringmaster Top Hat and Gold Monocle: The Meerkat Kingdom
Roman Armor: The Darkness
Roman Helmet: Skulldozer
White Neon Eyes: The Frozen Tundra
Wooden Sword: The Wedding Reception
Yellow Head: The Collector

Successfully complete one of the following tasks to get a trophy:

The Gardens (Bronze): Complete all levels in The Gardens.
The Savannah (Bronze): Complete all levels in The Savannah.
The Wedding (Bronze): Complete all levels in The Wedding.
The Canyons (Bronze): Complete all levels in The Canyons.
The Metropolis (Bronze): Complete all levels in The Metropolis.
The Islands (Bronze): Complete all levels in The Islands.
The Temples (Bronze): Complete all levels in The Temples.
Expert Creator (Bronze): Complete all levels in the Tutorials.
Artist (Bronze): Place a sticker.
Homemaker (Bronze): Place 10 stickers or decorations in your pod.
Fashion Sense (Bronze): Choose a costume for your sackperson with at least one item on your head, at least one item on your body, and a material.
Forager (Bronze): Collect 25% of the prize bubbles on the story levels.
Sticky Fingers (Bronze): Collect 50% of the prize bubbles on the story levels.
Treasure Hunter (Bronze): Collect 75% of the prize bubbles on the story levels.
2X Multiplier! (Bronze): Get a 2X Multiplier.
8X Multiplier! (Bronze): Get an 8X Multiplier .
Incredible Speed! (Bronze): Travel at incredible speed.
Incredible Height! (Bronze): Travel to an incredible height.
Friendly (Bronze): Complete a level with more than one player .
Party Person (Bronze): Complete a level online with 3 other players who are on your friends list.
Socialite (Bronze): Complete a level online with 3 other players who are not on your friends list.
Top of the class (Bronze): Win a 4-player game.
Traveller (Bronze): Complete a community leve.
FIRST! (Bronze): Be among the first 10 people to complete a community level.
Opinionated (Bronze): Tag a community level.
Neighbourhood (Bronze): Watch Heart 5 community levels.
Networking (Bronze): Heart 3 authors.
Talkative (Bronze): Post a comment on a community level.
Creator (Bronze): Build and save a level with the thermometer more than 30% full.
Team Creator (Bronze): Build and save a level with more than one player with the thermometer more than 30% full.
Publisher (Bronze): Publish a level.
Look what I made! (Bronze): A level you published was played by 5 or more people.
Just Beginning (Silver): Complete all the main path story levels.
20X Multiplier! (Silver): Get a 20X Multiplier.
Hi Score (Silver): Collect 1,000,000 points over all levels you have played.
Booty Master (Silver): Collect all the prize bubbles on the story levels.
Dr Frankenstein (Silver): Create a living creature with at least 2 eyes, 2 legs or wheels, and a brain.
Crowd Pleaser (Silver): A level you published was played by 50 or more people.
Feel the love (Silver): A level you published was hearted by 10 or more people.
Celebrity (Silver): You were hearted by 5 or more people as a player.
Play (Gold): Complete all story levels without dying, with the exception of those levels that only end when you die.
Create (Gold): A level you published was hearted by 50 or more people and you were hearted by 30 or more people.
Share (Gold): Play 150 community levels, tag 50 community levels and heart 10 community levels.
100% Complete (Platinum): Earn all LittleBigPlanet trophies to unlock this platinum trophy.

Additionally, there are four secret trophies:

Trendsetter (Bronze): Place a sticker or a decoration on another player's sackperson.
Cranium Collector (Bronze): Kill 100 Creatures across all levels.
Secret Stickerist (Bronze): Unlock the race in First Steps.
Sackbird (Bronze): Spend 8 seconds or more in the air.