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.