In this thesis we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of the universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locate local bifurcations in terms of these parameters. Here, we apply the proposed approach on Hopf-zero singularities whose the first few low degree terms are incompressible and generalized cusp case of Bogdanov-Takenz singular systems. In this direction, we obtain novel orbital and parametric normal form results. Moreover, we give a truncated universal asymptotic unfolding normal form and prove the finite determinacy of the steady-state bifurcations for two most generic subfamilies. We analyze the local bifurcations of equilibria, limit cycles and the secondary Hopf bifurcation of invariant tori.