  We study the simplest singular points of Fredholm maps of index zero between
Banach spaces, i.e. when the kernel of the Frechet derivative of the map has
dimension one. Even in this relatively simple case we have a rich variety of
singularities which are completely classified under the natural geometric
assumption of transversality of the map. In fact we have, locally, a suitable
stratification of the singular points that allows us to identify three kinds of
singularities: a) the ordinary or k-singularities (the infinite-dimensional
analogues of the well-known Morin singularities) and two new types, b) the
maximal-transverse singularities; c) the infinite-transverse singularities (the
latter ones can only occur in infinite dimensions).
