A good example is Sansa Stark (come on - are you really at all surprised I pulled out the Game of Thrones reference?). In a lot of ways her character falls into the damsel in distress trope [spoilers for A Song of Ice and Fire through A Storm of Swords, just FYI). Sansa is a young, beautiful girl who puts a lot of stock in the feminine courtly ideal - at the beginning of her narrative all she cares about is marrying the beautiful prince and having beautiful babies, and thriving in a court environment. Her world view is simplistic, and her understanding of the gray the world contains nonexistent. As far as she's concerned there is evil, and there are shining knights, and the knights always win. There is no other possible outcome.
But as her story continues she learns, she grows, and she begins to understand. More importantly, and perhaps more relevant, is she understands who she is as a young woman in an environment that only values her for her sexuality and her ability to produce children. She understands that she is valuable (and therefore able to remain alive) only so long as she fits into what the court wants to see: a demure, feminine, malleable young woman. She must appear stupid in order to remain alive. And she gets a lot of criticism for that.
But her entire narrative is about subverting the damsel in distress and twisting our expectations. Sansa is not stupid. Far from it and her internal narrative displays that - she's quite aware that she's in a den of lions and a single misstep will mean her death. She uses her position as a perceived young and stupid girl to keep living. She understands that she can't fight in a masculine way (those that do inevitably die without exception), but she's punished for playing into the feminine ideal. Her whole narrative is about criticizing the conditions she's forced to exist in - she can't ape her perceived betters (men) and she can't perform to their standard (the feminine ideal) without incurring wrath and risking death.
And understanding all of that produces a much richer writing and reading experience. Tropes are tropes for a reason: they're fun and we love them. But manipulating those tropes is always better because more often than not, you're going to create something better and your audience will get something much richer.