This paper contains some results on the geometrical properties of bifurcation varieties of sets and bifurcation varieties of functions for the simple singularities on manifolds with boundary. Let \(f(x):({\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)\) be a function with critical point 0 and \(F(x,\lambda)=f(x)+\sum\lambda_ i\phi_ i(x), i=0,...,\mu -1\), \(\phi_ 0\equiv 1\) a versal deformation of this function. The set of values \(\Sigma\) of parameters \(\lambda\in {\mathbb{C}}^{\mu}\) for which 0 is a critical value of function \(F(\circ,\lambda)\) is called the bifurcation variety (diagram) of sets of singularity f(x). The set of values \(\Xi\) of parameters \(\lambda '=(\ell_ 1,...,\lambda_{\mu -1})\in {\mathbb{C}}^{\mu -1}\) for which \(F(\circ,(0,\lambda '))\) has non-Morse critical points or multiple critical values is called the bifurcation variety of functions. The main results of this paper are the following. A theorem is proved that describes the covering map of the complement of the bifurcation variety of functions of simple singularity \({\mathbb{C}}^{\mu -1}-\Xi\) onto K\((Br(\mu)\),1), where B\(r(\mu)\) is a braid group with \(\mu\) strings. Also the author gives the normal form of vector field germ on \({\mathbb{C}}^{\mu}\supset\Sigma \) for the action of the group of diffeomorphism germs which preserves bifurcation variety \(\Sigma\). The second part of the paper describes the stratification of the bifurcation variety of functions \(\Xi\) in the terms of decompositions of singularities in their deformations.