  Using the principle of least action, the motion equations for a singular
hypersurface of arbitrary type in quadratic gravity are derived. Equations
containing the "external pressure" and the "external flow" components of the
surface energy-momentum tensor together with the Lichnerowicz conditions serve
to find the hypersurface itself, while the remaining ones define arbitrary
functions that arise due to the implicit presence of the delta function
derivative. It turns out that neither double layers nor thin shells exist for
the quadratic Gauss-Bonnet term. It is shown that there is no "external
pressure" for null singular hypersurfaces. The Lichnerowicz conditions imply
the continuity of the scalar curvature in the case of spherically symmetric
null singular hypersurfaces. These hypersurfaces must be thin shells if the
Lichnerowicz conditions are necessary. It is shown that for this particular
case the Lichnerowicz conditions can be completely removed therefore a
spherically symmetric null double layer exists. Spherically symmetric null
singular hypersurfaces in conformal gravity are explored as application.
