Kura Puzzles are unlocked the same way Wallpapers are, which is when certain events are triggered. Here is a list of all the puzzles you can currently unlock.
#1- Automatically obtained when you start Yume 2kki
#2 - Painter-kun's Painting - Interact with Painter-kun in the Art Gallery
#3 - Sign - When you first enter Sign World
#4 - Green Juice - Interact with Aojiru ("Cripple-Tan") at the Hospital
#5 - Three ┏（∵）┛ - When you first enter the Bowling Zone
#6 - Square - Wake up after your money is at least 800
#7 - もっきゅっ！ - Interact with the blob NPCs in the Marijuana Goddess World
#8 - Grinning Eyes - After you go up the ladder to Tapir-san's Place
#9 - Blue People - Unlock Wallpaper #97
#10 - In Vain - When you get a certain number of effects
#11 - Insect - When you get the Bug effect
#12 - Hakoko & Megusuri Uri - Interact with Hakoko at the Eyeball Shop
#13 - Darkness - When you go through the tunnel on the Highway
#14 - Pipe Organ - Interact with the Blue Beret Twin
#15 - Director's House -
#16 - Octopus - When you use the Fairy effect on Tako Otoko
#17 - Bane-kun - When you get the Spring effect
#18 - Odoriko-san - Interact with Odorika (you don't need the Maiko effect)
#19 - Gakuran - Interact with Gakuran-kun
#20 - Stardust - When you enter Space from Tapir-san's Place
#21 - Rune Mark - When you first enter Theater World
#22 - Primitive Present - When you first enter the Tribe Settlement
#23 - Little Red Riding Hood & Apple - When you get the Red Riding Hood effect
#24 - The Grudge - Cutting the rope at the well in Fairy Tale Woods and getting captured
#25 - Hanging Child - When you encounter Koshukei-tan (Hanged Girl)
#26 - Are You Going Again? - Interact with Megusuri Uri outside his shop in Visine World
#27 - The Playing Is... - Interact with the clown that plays the accordian at the Underwater Amusement Park
#28 - Mask - When you chainsaw Painter-kun
#29 - Penguin - When you get the Penguin effect
#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
#32 - Urotsuki_RT - When you get the Chainsaw effect
#33 - Smile - When you first enter Guts World
#34 - End - When you pass by the meteor along Highway (near the tunnel)
#35 - Would You Like To Dream? - After you go to sleep 30 times
#36 - Elf - When you get the Fairy effect
#37 - Candy Girl - When you use the Cake effect on Sweets Musume
#38 - Tear - When you chainsaw a stick person in the Penguin Game
#39 - May I Eat? - When you use the Cake effect on Sweets Musume
#40 - My Name - When you spell "Urotsuki" in the Spelling Room
#41 - Bug - When entering Seishonen's room and it is glitched
#42 - RPG-7_Untitled - When you get the chasers to surround you in Urotsuki's Dream Apartments, then leave the room and get attacked
#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 - When you kill the angel priest in Fountain World
#49 - When you are chainsawwed by the hockey mask girl in GALAXY Town
#50 - After encountering Takofuusen in the Neon City.
#51 - When you have a cup of coffee at the cafe in the Eyeball Archives.
#52 - When you approach the pink structure in the Polluted Swamp
#53 - When you find the caged Shadow Bird in the Halloween Forest part of the Cat Cemetery.
#54 - Nigella - After you wake up when you have some money