For a Zariski general (regular) hypersurface V of degree M in the (M+1)-dimensional projective space, where Mgeqslant 16, with at most quadratic singularities of rank geqslant 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that V is non-rational and its groups of birational and biregular automorphisms coincide: mathrm{Bir} V = mathrm{Aut} V. The set of non-regular hypersurfaces has codimension at least frac{1}{2}(M-11)(M-10)-10 in the natural parameter space.