  We study the threshold behaviour of two dimensional Schr{\" o}dinger
operators with finitely many local point interactions. We show that the
resolvent can either be continuously extended up to the threshold, in which
case we say that the operator is of regular type, or it has singularities
associated with $s$ or p-wave resonances or even with an embedded eigenvalue at
zero, for whose existence we give necessary and sufficient conditions. An
embedded eigenvalue at zero may appear only if we have at least three centres.
  When the operator is of regular type we prove that the wave operators are
bounded in $L^p(\R^2)$ for all $1<p<\infty$. With a single center we always are
in the regular type case.
