Even though most of us don't use horses or buggies as our main transportation these days, our language still has horse-related idioms, which writers sometimes mix up.
One idiom that came up in a story I edited this week was rendered "full reign," to imply that something was unhampered. In fact, the idiom is "full rein" as in allowing a horse to gallop all out. "Free rein" is a variation of that.
Sometimes, writers report that efforts have been made to "reign in" an activity. That, too, should be "rein," as when a rider pulls up on the horse's reins to slow or stop the animal.
Also, if a person is in power, he "holds the reins," not "reigns." On the other hand, if a person has power over a group, he "reigns supreme."