In this paper, we consider 3x3 matrices which represent the linear part of vector fields in \(\mathbb {R}^{3}\) that are tangent vector fields when restricted to one of the following singular subsets: a cone, a cross-cap, a double crossing or a triple crossing point (transversal intersection of two or three planes in general position, respectively). We present a dynamical classification in the spirit of the trace-determinant plane for each of these singular subsets. Moreover, we propose a slightly different type of continuation, where we study the bifurcations that occur when we consider one-parameter families of matrices that make the transition from a vector field tangent to one singular subset to another. These transitions occur naturally in vector fields tangent to closed singular two-manifolds with the aforementioned singular sets, called Gutierrez-Sotomayor surfaces. Through the analysis of these continuations we relate qualitative changes on the dynamics to the geometry of the singular subsets.
