  We prove birational superrigidity of every hypersurface of degree N in P^N
with singular locus of dimension s, under the assumption that N is at least
2s+8 and it has only quadratic singularities of rank at least N-s. Combined
with the results of I. A. Chel'tsov and T. de Fernex, this completes the list
of birationally superrigid singular hypersurfaces with only ordinary double
points except in dimension 4 and 6. Further we impose an additional condition
on the base locus of a birational map to a Mori fiber space. Then we prove
conditional birational superrigidity of certain smooth Fano hypersurfaces of
index larger or equal to 2, and birational superrigidity of smooth Fano
complete intersections of index 1 in weak form.
