  Effective methods are introduced for testing zero-dimensionality of varieties
at a point. The motivation of this paper is to compute and analyze deformations
of isolated hypersurface singularities. As an application, methods for
computing local dimensions are also described. For the case where a given ideal
contains parameters, the proposed algorithms can output in particular a
decomposition of a parameter space into strata according to the local dimension
at a point of the associated varieties. The key of the proposed algorithms is
the use of the notion of comprehensive Gr\"obner systems.
