Complete the indicated task to unlock the corresponding dungeon.

Amber Garden: Unlock Charme. View the event in the town square of Tielle chasing after her sister. Charme will enter into your store and mention a new dungeon.
Crystal Nightmare: Complete Lapis Ruins.
Jade Way: Complete Hall of Trials
Lapis Ruins: Pay off the debt and unlock every character except Arma. Arma will enter your store and unlock this dungeon.
Obsidian Tower: Defeat Tielle in Amber Garden, then watch both night events in the town square with Griff. Have your store towards a dark atmosphere to get Griff to come in during one of the two last slices of time. The next day Louie will enter your store and mention the new dungeon.