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We study the simplest singular points of Fredholm maps of index zero betweenBanach spaces, i.e. when the kernel of the Frechet derivative of the map hasdimension one. Even in this relatively simple case we have a rich variety ofsingularities which are completely classified under the natural geometricassumption of transversality of the map. In fact we have, locally, a suitablestratification of the singular points that allows us to identify three kinds ofsingularities: a) the ordinary or k-singularities (the infinite-dimensionalanalogues of the well-known Morin singularities) and two new types, b) themaximal-transverse singularities; c) the infinite-transverse singularities (thelatter ones can only occur in infinite dimensions).

Singularities of Fredholm maps with one-dimensional kernels, I: A complete classification