The paper is concerned with the classification of typical singularities for a class of four-dimensional discontinuous vector fields. Being inspired by the work of D. V. Anosov [Autom. Remote Control 20, 130–143 (1959; Zbl 0093.09204)], the authors consider the semilinear vector field \(L_{a},\) \(a\in\mathbb{R},\) \[ x_{2}\frac{\partial}{\partial x_{1}}+x_{3}\frac{\partial}{\partial x_{2} }+x_{4}\frac{\partial}{\partial x_{3}}+a^* \operatorname{sgn} \left( x_{1}\right) \frac{\partial}{\partial x_{4}} \] and examine the vector fields which are deformations of \(L_{a}\) preserving some distinguished properties of the unperturbed system. A typical example is the set \(W\) of vector fields \(Z_{a,\lambda,\mu},\) \(a,\lambda,\mu\in \mathbb{R},\) defined by \[ \left( x_{2}+\lambda x_{4}^{2}+\mu x_{4}^{3}\right) \frac{\partial}{\partial x_{1}}+x_{3}\frac{\partial}{\partial x_{2}}+x_{4}\frac{\partial}{\partial x_{3}}+a^* \operatorname{sgn} \left( x_{1}\right) \frac{\partial}{\partial x_{4}} \] and termed the standard (reversible) model. The authors start with the discussion of reversibility properties of the system and proceed with a complete classification of reversible singular vector fields (Theorem 1). Making use of the local models for regular and singular points, they derive a sufficient condition for the uniqueness of solutions for such vector fields and demonstrate that generic typical singularities fill up a two-dimensional manifold of \(H_{0}=\{x_{1}=0\}.\) Then bifurcation diagrams are exhibited and the stratification of the singularities is discussed. The paper concludes with the analysis of the behavior of some classes of nonreversible vector fields and nongeneric vector fields.