This paper investigates qualitative features of singular points of differential-algebraic power system model. Large-scale dynamical systems such as power systems are generally modeled as a set of parameter dependent differential-algebraic equations (DAE's). As the parameter changes, the DAE model of a power system may exhibit a singularity-induced (SI) bifurcation as well as other frequently encountered bifurcations such as saddle node and Hopf types. The SI bifurcation point, which occurs when system equilibria meet the singularity of the algebraic part of the model, belongs to a large set of other singular points call ed a singular set. At any singular point in this set, the DAE model breaks down, meaning a vector field on a set satisfying algebraic part of the DAE model (known as constraint manifold) is not well defined. Thus, DAE's cannot be reduced to a set of ordinary differential equations (ODE's) at the singular points and ODE solvers fail to converge in a neighborhood of a singular point. The singular points split into two complementary classes: Those from which exactly two distinct trajectories emanate (repelling) and those at which exactly two distinct solutions terminate (attracting). In order to predict system trajectories originating from a point close to a singular point, the qualitative properties of singular points at any given parameter value are identified and simulation results are presented for a 3 bus test system.