  Inverse scattering theory is extended to one-dimensional Schr\"odinger
problems with near-boundary singularities of the form $v(z\to 0)\simeq
-z^{-2}/4+v_{-1}z^{-1}$. Trace formulae relating the boundary value $v_0$ of
the nonsingular part of the potential to spectral data are derived. Their
potential is illustrated by applying them to a number of Schr\"odinger problems
with singular potentials.
