Running away from home. Doing something that would get them abducted by the villain of the piece (though that's usually off-stage, because then she's the distressed damsel.) Searching for or hiding from the hero of the story during a challenge for her hand, though that one usually involves magic. The closest I've seen is that sometimes when a man appears in her bedroom, he has to vanish when her panicked reaction rouses them, until they are convinced it's nothing.
The problem is that this won't really work in many fairy-tale retellings. Like one where the prince is disguised as a gardener's boy and the princess suspects it, and is trying to find out. But does realize that the ladies-in-waiting would gossip no end. . . going to have to justify how she manages to get him alone enough to find out without producing gossip from that.