  We focus on topological equisingularity of families of holomorphic function
germs with 1-dimensional critical set. We introduce the notion of
equisingularity at the critical set and prove that any family which is
equisingular at the critical set is topologically equisingular. We show that if
a family of germs with 1-dimensional critical set has constant generic Le
numbers then it is equisingular at the critical set, and hence topologically
equisingular (answering a question of D. Massey). We use this to modify the
definition of singularity stem present in the literature, introducing and
characterising topological stems (being this concept closely related with
Arnold's series of singularities). We provide another sufficient condition for
topological equisingularity for families whose reduced critical set is deformed
flatly. Finally we study how the critical set can be deformed in a
topologically equisingular family and provide examples of topologically
equisingular families whose critical set is a non-flat deformation with
singular special fibre and smooth generic fibre.
