A common sexist trope from fairy tales is the evil stepmother, as seen in "Cinderella" and "Snow White". Another similar trope is the evil witch, as seen in "Sleeping Beauty", "The Little Mermaid" and "Rapunzel". These two tropes ultimately imply that women are a threat to one another and will not support each other. The existence of these tropes means that the female protagonist often has no female role model and relies on men for positive relationships. These tropes also combine sexism with ageism - suggesting that older women will become deathly jealous of younger women, and will end up ugly and alone.
One very concerning sexist trope in fairy tales is romanticized sexual violence. In Giambattista Basile's "Sun, Moon, and Talia", the original "Sleeping Beauty", the princess only wakes up when she is giving birth to the prince's children because she was raped in her sleep. In the original "Snow White", the princess is kidnapped by the prince while she is unconscious.
While we see a less vivid depiction of this sexual violence in the Disney versions, it is still alluded to and female characters do not consent to be kissed or touched.
The final trope worth mentioning is the 'knight in shining armor', a trope carried forward from medieval ideas about chivalry and courtly love. The princess or female character is often painted as a victim who lives an uneventful life alone. They are then rescued by a prince, married, and live happily ever after. This celebration of matrimony implies that women who are unmarried are worthless, and this idea is outdated in most modern societies.
Since fairy tales are meant to teach morality, they should be amended to contain messages that actually empower girls today. Otherwise, they have no use to society anymore. Some examples of modern retellings of fairy tales that send good messages to women are "The Lunar Chronicles" by Marissa Meyer and "The Bloody Chamber" by Angela Carter.