In the present work, we propose a numerical analysis of instability and bifurcations for geometrically nonlinear elasticity problems. These latter are solved by using the Asymptotic Numerical Method (ANM) associated with the Method of Fundamental Solutions (MFS). To compute bifurcation points and to determine the critical loads, we propose three techniques. The first one is based on a geometrical indicator obtained by analyzing the Taylor series. The second one exploits the properties of the Padé approximants, and the last technique uses an analytical bifurcation indicator. Numerical examples are studied to show the efficiency and the reliability of the proposed algorithms.