  We prove birational superrigidity of Fano double hypersurfaces of index one
with quadratic and multi-quadratic singularities, satisfying certain regularity
conditions, and give an effective explicit lower bound for the codimension of
the set of non-rigid varieties in the natural parameter space of the family.
The lower bound is quadratic in the dimension of the variety. The proof is
based on the techniques of hypertangent divisors combined with the recently
discovered $4n^2$-inequality for complete intersection singularities.
