Here is the full list of all 36 Besiege achievements.
Conquer Ipsilon
Travel at speeds faster than is sensible
Set yourself on fire
Complete a campaign level under 2 seconds
Complete a campaign level with under 3 blocks
Build your first level using the level editor
Share your machine with the whole world
Subscribe and download someone else's machine
Get hit by lightning
Put out a burning block using water
Defeat 1000 AI units
Beat Queens Fodder without using fire or explosives
Beat Scouts of Tolbrynd without cannons, flamethrowers or crossbows
Steal the Duke's knowledge in under 2 minutes!
Ring all 3 Awakening Bells at the same time
Beat the Consumed King without using mechanical blocks
Complete Standing Stone without using armor
Beat the Southern Shrine by primarily using fire to kill the birds
Destroy the Duke's freighters without flying blocks
Beat Old Howl Battlefield while having blown up all the bombs
Conquer Tolbrynd
Conquer Valfross
Complete Perimeter Wall without pressing a key
Complete Old Mining Site without pressing a key
At least 2 players joined your server
Join over 10 servers
Complete the Mountain Barrier without explosives
Complete the Revolving Monolith without using Grabbers
Complete Towering Eye without ever being hit
Complete Tree of Akhmora without using Crossbows, Cannons and Explosives
Lift the boulder in Stock Tower 10 units off the ground
Complete Krolmar
Complete the Campaign
Complete a player-made map
Complete Kahraz Village without using Cannons, Crossbows, Drills or Saws
Reach enough altitude to freeze your machine