  We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety
with respect to equivariant special test configurations, in terms of its moment
polytope and some combinatorial data associated to the open orbit. Combined
with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano
manifolds proved by Datar and Sz\'ekelyhidi, it yields a criterion for the
existence of a K\"ahler-Einstein metric on a spherical Fano manifold. The
results hold also for modified K-stability and existence of K\"ahler-Ricci
solitons.
