In Star Wars: Episode III – Revenge of the Sith, Dooku fights Kenobi and Anakin in front of the 'captured' Palpatine. At the end of the duel, Anakin disarms (and dis-hands) Dooku and takes his lightsaber (and uses it, along with his own saber, to decapitate Dooku). Anakin then 'pilots' the ship to Coruscant, presumably still in possession of Dooku's saber ...
What happens to Dooku's lightsaber after that?
Canon answers are preferred. A 'best' answer will probably include both Canon and Legends answers.