The Red Hat Society is a women's social organization that was originally just for women 50 and older, but is now open to women of all ages. The name of the group comes from the fact that members who are 50 and older wear the signature red hats (with purple outfits) while members who are under fifty wear pink hats (with lavender-colored outfits).
They were parodied in “The Last of the Red Hat Mamas" the seventh episode of the seventeenth season of The Simpsons. In the episode, Marge befriends a group of women called the Cheery Red Tomatoes, who turn out to be a gang of thieves.
The Shriners stand out as being an exceptionally (and intentionally) silly hat wearing society, with the fezzes and little cars and all.
They also liked Ray Stevens' song "Shriner Convention" (which is basically about drunken more-or-less upstanding middle-aged hijinx) enough to book him to play it at actual conventions.
Believe it or not, The Illuminati were basically what happened when a number of eccentric scholars and philosophers (read: 18th-century nerds) decided to form a hat (and robes, and masks…) wearing society. They tried to become an Ancient Conspiracy, but how well they succeeded depends on which Conspiracy Theorists you listen to.