The main result of the article is that if a hypersurface \(F\) of degree \(d\) in the complex projective \(n\)-space, \(n\geq 2\), has only isolated singularities and satisfies \(\tau(F)<4d-4\), where \(\tau(F)\) is the total Tjurina number of \(F\), then the germ at \(F\) of the space of hypersurfaces of degree \(d\) is a versal deformation of the multisingularity of \(F\), and the germ of the equisingular family at \(F\) of hypersurfaces of degree \(d\) is smooth. For the case of semiquasihomogeneous singular points we give a similar sufficient condition for the smoothness of the \(\mu\)=const strata of hypersurfaces of the given degree. This is an extension of known results for plane curves to higher dimensions.