Kura Puzzles are unlocked the same way Wallpapers are, and in most cases, they are linked: always check your wallpapers after encountering characters or triggering events mentioned below. Here is a list of all 57 puzzles you can currently unlock. You can press Shift for a hint.
#2 - Painter-kun's Painting - Interact with Painter-kun in the Art Gallery.
#5 - Three ┏（∵）┛ - When you first enter the Bowling Pin World from Bowling Zone.
#8 - Grinning Eyes - After you go up the ladder to Tapir-San's Place.
#9 - Blue People - Unlock Wallpaper #74.
#11 - Insect - When you get the Insect effect.
#12 - Hakoko & Megusuri Uri - Interact with Hakoko at the Eyeball Shop.
#15 - Director's House - Unlock Wallpapers #64, #85, and #88.
#17 - Bane-kun - When you unlock Wallpaper #36 after getting the Spring effect.
#19 - Gakuran - Interact with Gakuran-kun.
#21 - Rune Mark - When you first enter Theatre World.
#22 - Primitive Present - When you first enter the Tribe Settlement.
#24 - The Grudge - Cutting the rope at the well in Fairy Tale Woods and getting captured.
#27 - The Playing Is... - Interact with the clown that plays the accordion at the Underwater Amusement Park.
#30 - Black Witch - After chainsawing the sane Shadow Woman in Forest World, going underground and then emerging.
#31 - Eyeball Human - When you encounter the Eyeball People in the Spelling Room.
#33 - Smile - When you first enter Guts World.
#34 - End - When you pass by the meteor along Highway (near the tunnel).
#38 - Tear - When you chainsaw a stick person in the Penguin Game.
#40 - My Name - When you spell "Urotsuki" in the Spelling Room.
#41 - Bug - When entering Seishonen's room when it's glitched.
#43 - Uro & T-Shirt-kun - Interact with the blue NPC at the end of the Arrow Maze.
#44 - Room Window - When you go through the black window in the Never-Ending Hallway.
#45 - Each World - When you go through all of the Nexus portals.
#46 - NULL MI - After looking up at the ceiling of the inescapable room in Binary World's tower.
#47 - Idenshideau - After meeting the fish in the jar in Green Binary World.
#48 - ぜんまいの鶏 - When you kill the angel priest in Fountain World
#49 - ジェイ子 - When you are chainsawed by the hockey mask girl in GALAXY Town
#50 - 暗彩光を飛ぶ - Unlocked by meeting "It Came From Behind" in Neon City.
#51 - Coffee Urotsuki - When you have a cup of coffee at the cafe in the Eyeball Archives.
#52 - 退水の先に - When you approach the pink structure in the Polluted Swamp
#53 - Within the Cage - When you find the caged Shadow Bird in the Halloween Forest part of the Cat Cemetery.
#55 - 揺蕩うくじら - When you enter Paradise for the first time.
#56 - When you get a ticket from the ticket dispenser in the Overgrown Condominium's arcade room.
#57 - After entering the head with the smiling creature in Head World.