  We look at topological equisingularity of a holomorphic family of reduced
mapping germs f_t:(C^3,O)->C over a contractible base T having non-isolated
singularities, by means of their normalisations. We introduce the notion of
Equisingularity at the Normalisation for a family f_t and prove that, in many
cases, it characterises topological embedded equisingularity and
$R$-equisingularity. Moreover we apply our results to the study of topological
A-equisingularity of parametrised surfaces, and in many cases characterise it
in terms of the constancy of the Milnor number of the inverse image of the
singular sets of the parametrised surfaces. A novelty of our approach is that
our topological trivialisations are global in the base.
