  We prove Manin's conjecture on the asymptotic behavior of the number of
rational points of bounded anticanonical height for a spherical threefold with
canonical singularities and two infinite families of spherical threefolds with
log terminal singularities. Moreover, we show that one of these families does
not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in
the asymptotic formula. Our proofs are based on the universal torsor method,
using Brion's description of Cox rings of spherical varieties.
