This is probably the most commonly used by modern authors: the idea is to take advantage of a plot that everyone is already familiar with and to surprise the reader by twisting one of the assumed elements of the plot. This can be done on several different levels.
In a traditional folk tale, things are pretty black and white. There's good and there's evil-that which is good is all good, from every action to their flawless appearance, and likewise that which is bad is evil through and through. Some stories take these ideas of black and white and totally turn them on their heads, such as Tanith Lee's short stories in her collection Red as Blood: Tales from the Sisters Grimmer. In this book, Rapunzel and Snow White are evil and their stepmothers innocent victims of their Satanic activities. This kind of twisted fairy tale can be the most shocking, but can become predictable if overdone (Tanith Lee, in my opinion, does a very good job of twisting tales into very dark retellings but maintaining quality stories. This is very hard to do). But these worlds are still largely seen in black and white.
Some versions will take be faithful in most elements of the tale but take one and twist it. In The Fairies Return, Robert Speaight's Prince Charming turns out to be less than honorable and Cinderella learns that a happy ending does not necessarily need a marriage in it (can I hear an 'Amen'?). Many modern versions of Beauty and the Beast end with Beauty embracing her animal side more so than the Beast conforming to the cultural terms of a gentleman; many modern versions of Sleeping Beauty interpret the pricking of the spindle as having sex or injecting herself with drugs (the latter also found in The Fairies Return-G.B. Sterns' idea predates Francesca Lia Block's by about 70 years).