  We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z)
= \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 -
\operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow
\mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear
regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z)
= \dfrac{1 + \varphi(F/\varepsilon)}{2}X(z) +\dfrac{1 - \varphi(F
/\varepsilon)}{2}Y(z),$$where $\varphi$ is a transition function (not
necessarily monotonic) and nonlinear regularizations of the vector field $Z$
whose transition function is monotonic. It is a well-known fact that the
regularized system is a slow-fast system. The main contribution of this paper
is the study of typical singularities of slow-fast systems that arise from
(linear or nonlinear) regularizations. We developed an algorithm to construct
suitable transition functions, and we apply these ideas in order to create
slow-fast singularities from normal forms of piecewise smooth vector fields. We
present examples of transition functions that, after regularization of a PSVF
normal form, generate normally hyperbolic, fold, transcritical, and pitchfork
singularities.
