The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Grušin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study propagation of the singularities. The results are expressed in terms of the real values of a complex function extending the Carnot-Caratheodory distance, and the geodesics of the associated sub-Riemannian geometry play a crucial role in the analysis.