Although Finn has an unsuccessful crack at Anakin's old lightsaber, Rey does finally claim it for herself at the end of The Force Awakens. After using it to defeat and scar Kylo Ren, the blue blade becomes Rey's primary weapon through the Star Wars sequels.

Whose lightsaber is Rey using?

As I mentioned, Rey buries Luke and Leia's twin sabers in the sand and then she gets out something new: her own lightsaber with a gleaming golden blade. Up until now, Rey has mostly been using Anakin's old blue lightsaber—just as Luke did for a time before her.

Who's lightsaber does Rey have at the end of rise of Skywalker?

At the end of the film, Rey buries Luke and Leia's lightsabers on Tatooine. (As a side note: It's great that Leia did end up doing her Jedi training and had her own lightsaber.