  Solutions to the stochastic wave equation on the unit sphere are approximated
by spectral methods. Strong, weak, and almost sure convergence rates for the
proposed numerical schemes are provided and shown to depend only on the
smoothness of the driving noise and the initial conditions. Numerical
experiments confirm the theoretical rates. The developed numerical method is
extended to stochastic wave equations on higher-dimensional spheres and to the
free stochastic Schr\"odinger equation on the unit sphere.
