We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase and/or initial amplitude. To estimate quantities of interest related to the solution, and their statistics, we combine a high frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution  is highly oscillatory in both physical and stochastic space, we show theoretical arguments and numerical evidence that quantities of interest based on local averages of  are smooth, with derivatives in stochastic space uniformly bounded in frequency. The stochastic regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We show numerical tests that demonstrate this advantage.