Here is the full list of all 36 Don't Starve Together: Console Edition trophies - there are 20 bronze, 10 silver, 5 gold and 1 platinum.
Found a hole in the ground. Wow!
Willingly jumped down a gross hole.
You filled your trophy case to brimming.
Survived 20 consecutive days.
Survived 35 consecutive days.
Survived 55 consecutive days.
Survived 70 consecutive days.
Built a Science Machine.
Built an Alchemy Engine.
Built a Shadow Manipulator.
Built a Prestihatinator.
Struck a deal with a large porky monarch.
Grew a pretty flower. Aw.
Harvested some sweet, sweet honey from an overflowing beebox.
Demonstrated your formidable sewing skills.
Had six simultaneous pigman followers. What a party!
Had four simultaneous rock lobster followers. Snappy!
Witnessed the miracle of Tallbird motherhood.
Soothed an enraged tree.
You felt the earth move under your feet and the sky tumblin' down.
Travelled in the exact opposite of style.
Wore Snurtle Shell Armor and a Shelmet at the same time.
Visited a public world.
Hosted a public world for 40 consecutive days.
Brought a friend back from the dead.
Give resources to a new player after establishing yourself for 20 days.
Showed everyone your true feelings with an emote.
Donned a fashionable character skin.
Crafted a new clothing item with the help of a mysterious Innkeeper.
Defeated a Deerclops.
Defeated a Spider Queen.
Defeated an Ancient Guardian.
Defeated a Moose. Or a Goose. Or both?
Defeated a Bearger.
Defeated a Dragonfly.
Domesticated a wild beefalo.