What happened to Joseph Smith’s seer stone? Does the church have one today?
Joseph Smith owned several seer stones throughout his life. From what historians can gather, it seems that he gave the stone that was used during translation of part of the Book of Mormon to Oliver Cowdery. After Cowdery’s death, Brigham Young’s brother Phineas happened to visit with Elizabeth Cowdery, Oliver’s widow and sister of David Whitmer. Elizabeth gave the stone to Phineas, who brought it back to Salt Lake and turned it over to President Young. That stone remains in possession of the Church, having apparently been publicly displayed at the dedication of the Manti Temple in 1888 and having been mentioned by Elder Joseph Fielding Smith in his Doctrines of Salvation, published around 1954.
At least one other stone that the Prophet owned at one point, was given to an early Church member named Philo Dibble and remains in private hands. I understand the Church also owns one or two other seer stones also used by Joseph Smith at various times in his life, though it has not publicly said very much about these sacred objects.