  We study boundary singularities which can appear for infinitesimal generators
of one-parameter semigroups of holomorphic self-maps in the unit disc. We
introduce "regular" fractional singularities and characterize them in terms of
the behavior of the associated semigroups and Koenigs functions. We also
provide necessary and sufficient geometric criteria on the shape of the image
of the Koenigs function for having such singularities. In order to do this, we
study contact points of semigroups and prove that any contact (not fixed) point
of a one-parameter semigroup corresponds to a maximal arc on the boundary to
which the associated infinitesimal generator extends holomorphically as a
vector field tangent to this arc.
