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They have a a church dedicated to them called the Grand Cathedral (which is also a fighting stage in the game). Many people come to the church to give their confessions, tithes, and so on. Because the Trinity are females, people will say "Goddess-speed" to anyone who could use luck.
However, they are the ones who created the Skull Heart, hoping to turn the ladies of the Canopy Kingdom into Skullgirls, so they could use them to destroy the world. Marie, however, has a strong heart, so she couldn't be controlled by the Skull Heart, leaving her to continue her own mission to destroy the Medici Mafia for messing with her and Peacock. The Trinity did not accept this and wanted Double to take her down. Parasoul and Double worship them with their inverted cross necklaces. Double calls them "masters".
In Valentine's story, she intends to put an end to them and their plans of destruction. She wishes to become a Skullgirl not only to atone for turning Carol into Painwheel, but also to get a better look at them. She is hoping that her research will help the Labs in the future to bring them down for good.
It is unknown as to why The Trinity wants to destroy the world.