Shiren Post-Story
1. Inori Cave - After beating the main game, it will be accessible in the Starter's House (It's the house where you can take tutorial stages).
2. Lost Well - Talk to the Sparrow in the lower left area of the Sparrow Inn (this is located in Sparrow Village, the place reached before going to Miracle Tower 4F).
3. Gen's Turf - Talk to the old lady in Inori Village (second house from the left) and you should receive a Catnip Juice. Bring that to Gen (the cat man). Go clear a dungeon and come back to his place and you will notice a shining pot.
4. Pitfall of Life - Reach the top of Fortune Tower with Kojirota in your party (he must not die along the way). When you return to the village, go to the Elder's House to see the event to unlock the dungeon.
5. Heavenly Lake - After beating the main story, talk to the old man with the boat in the lower right part of Nekomaneki Village.
6. Old Road - Level Kon and Gen to Level 10, then level up Koharu's Sword to 2nd rank. Once all those conditions are satisfied, go to the hotel to see an event. Afterwards. go to Gen's House and examine the challenge letter on the ground. To access the dungeon, go to the entrance of Inori Village.
7. Bizarre Tower - Reach the top of the Miracle Tower with Tao in your party (she must not die along the way). You also need to see the event with her in the Sparrow Inn and this will not occur if you bring the Sparrow with you along the way (it's a random NPC in the dungeon and he will ask you if you can escort him to the village, just don't take him). After beating the tower with her, go back to the entrance of the 3 towers and talk to Tao and she can escort you to the tower.
8. Onigiri Hollow - There are too many steps to unlock this dungeon. First, you need to talk to a girl in Nekomaneki Town (IIRC it's the girl just outside the hotel), then leave for a dungeon then come back and talk to her. Talk to the blonde guy in the upper left area of Nekomaneki Town. Then go to a dungeon and come back (you can simply just go to Destiny Road and use an Escape Scroll). You will basically have to talk to this guy (and do whatever he says) and then go to a dungeon then return to the village and talk to him again, repeat the same steps several times until you reach the end of the chain of event. Once you see the last event, the dungeon will be open to you.
9. Primordial Chasm - This is the second hardest dungeon in the game. After beating the game, go to the blacksmith's house in Hermit's Retreat for an event. After the event, the dungeon should now be accessible.
10. Destiny's Descent - This is the hardest dungeon in the game, where you start at level 1 and all items found are unidentified. Prepare to experience all the unfair abilities of the monsters in the game (like that rabbit that will teleport you (anywhere you are) and all the monsters in the floor to the same area, closely together to boot lol, or that dragon that can spew fire at you anywhere you are in the map). Anyway, to access the dungeon, gather 7 different cat stones, bring them to the old man in Inori Cave and make a wish. It should be the last option I believe (it says something about "I want to challenge a much difficult dungeon"). The entrance to the dungeon is in the entrance to the 3 towers.