  For a Zariski general (regular) hypersurface $V$ of degree $M$ in the
$(M+1)$-dimensional projective space, where $M$ is at least 16, with at most
quadratic singularities of rank at least 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. The set of non-regular
hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the natural
parameter space.
