The dynamics of a large class of physical systems such as the general power system can be represented by parameter-dependent differential-algebraic models of the form x = f and O = g. Typically, such constrained models have singularities, This paper analyzes the generic local bifurcations including those which are directly related to the singularity. The notion of a feasibility region is introduced and analyzed. It consists of all equilibrium states that can be reached quasistatically from the current operating point without loss of local stability, It is shown that generically loss of stability at the feasibility boundary is caused by one of three different local bifurcations, namely the well-known saddle-node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation which is analyzed precisely here for the first time. The latter results when an equilibrium point is at the singular surface, Under certain transversality conditions, the change in the eigenstructure of the system Jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed, It is show-a generically that the constrained system behavior becomes unpredictable at the singularity induced bifurcation, Global bifurcations do not directly affect stability of equilibria, and as such they cannot occur on the feasibility boundary, Physical phenomena connected with the bifurcations are analyzed with an eye on the limitations of the constrained model, The results presented in this paper are general and applicable to any differential-algebraic equations system, while being motivated here by the large power system.