Answers to some equisingularity questions

We give examples of families of hypersurface singularities with constant Le numbers, constant Milnor fibration and non-constant topological type, answering negatively a question of D. Massey. On the other hand we prove that the constancy of the Le numbers implies that the homotopy type of the link is constant along the family. As an application we give an example of a flat family of projective reduced and irreducible hypersurfaces having the same homotopy type but different topological type. Another application is an example of a Whitney-trivial family of isolated singularities such that the topological type of the projectivised tangent cone is not constant in the family. The last example answers negatively a question of O. Zariski.