Talk to Mystic the first time you reach Serendipity to get Mog's Manifestation. When you have this your Moggle's throw and Haggler skill will be improved and can be used at the indicated areas to get the corresponding unique monster crystal.
Cactuarama (A Dying World 700): Throw Mog into a position close to Chocolina in the Black Sands Zone.
Cactuarina (New Bodhum 003AF): Throw Mog into the crater beneath the gate in the Meteorite Impact Site Area.
Chichu (Archylte Steppe ???AF): Throw Mog into the patch of red flowers close to the Cactuar Waystone on The Plains Of Eternity.
Leyak (Sunleth Waterscape 300AF): Throw Mog at the two Miniflans while riding the giant beast in the Anime Trail Area.
Nanochu (Archylte Steppe ???AF): Throw Mog onto the small island at the center of the Clearwater Marshes.
Rangda (New Bodhum 003AF): Throw Mog into the ocean from the pier east of the NORA house.
Silver Chocobo (Academia 4XXAF): Throw Mog onto the platform beneath the holographic projection of Cocoon.