The authors consider a \(d\)-dimensional random field \(u = \{u(t,x)\}\) that solves a non-linear system of stochastic wave equations in spatial dimensions \(k \in \{1,2,3\}\), driven by a spatially homogeneous Gaussian noise that is white in time. They mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent \(\beta\). Using Malliavin calculus, they establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of \(\mathbb{R}^d\), in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when \(d(2-\beta) > 2(k+1)\), points are polar for \(u\). Conversely, in low dimensions \(d\), points are not polar. There is, however, an interval in which the question of polarity of points remains open.