There are 14 Trophies that can be earned in this title.

Bronze

Chef Supreme
Compose the perfect dish

King of Manhattan
Complete a Taxi level with 5 "Fast rides"

King of Catapult
Get 5 'Catapult Bonuses'

Elite Swinger
Get 8 'Great Swings" in a timed level

Femme Fatale
5 Flying pickups in one Level

Fishing ala Dundee
Catch a shark with an explosive

FAA Approved
Complete a Taxi level without taking any damage

King of the hill
Get 85+m at the ski jump

Cooking with style
Make a 25+ points Hoopbonus in the IronChef level

Squid Master
Beat the Squid without taking any damage

Golden cab
Complete a Gold medal taxi mission

Silver

Got the golds
Gained All Gold Medals at Difficulty Medium

Easy Peasy Gold
Gained All Gold Medals at Difficulty Easy

Gold

Hard earned Gold
Gained All Gold Medals at Difficulty Hard

http://www.ps3trophies.org/game/landit-bandit/trophies/