  The paper is devoted to regularity theory of generalized solutions to
semilinear wave equations with a small nonlinearity. The setting is the one of
Colombeau algebras of generalized functions. It is shown that in one space
dimension, an initial singularity at the origin propagates along the
characteristic lines emanating from the origin, as in the linear case. The
proof relies on a fixed point theorem in the ultra-metric topology on the
algebras involved. The paper takes up the initiating research of the 1970s on
anomalous singularities in classical solutions to semilinear hyperbolic
equations and transplants the methods into the Colombeau setting.
