  The retarded Green function of a wave equation on a 4-dimensional curved
background spacetime is a (generalized) function of two spacetime points and
diverges when these are connected by a null geodesic. The Hadamard form makes
explicit the form of this divergence but only when one of the points is in a
normal neighbourhood of the other point. In this paper we derive a
representation for the retarded Green function for a scalar field in
Schwarzschild spacetime which makes explicit its {\it complete} singularity
structure beyond the normal neighbourhood. We interpret this representation as
a sum of Hadamard forms, the summation being taken over the number of times the
null wavefront has passed through a caustic point: the sum of Hadamard forms
applies to the non-smooth contribution to the full Green function, not only the
singular contribution. (The term non-smooth applies modulo the
causality-generating step functions that must appear in the retarded Green
function.) The singularity structure is determined using two independent
approaches, one based on a Bessel function expansion of the Green function, and
another that exploits a link between the Green functions of Schwarzschild
spacetime and Pleba{\'n}ski-Hacyan spacetime (the latter approach also yields
another representation for the {\it full} Schwarzschild Green function, not
just for its non-smooth part). Our representation is not valid in a
neighbourhood of caustic points. We deal with these points by providing a
separate representation for the Green function in Schwarzschild spacetime which
makes explicit its (different) singularity structure at caustics of this
spacetime.
