In this paper, we are interested in the stability analysis of multilayer thin shells using asymptotic numerical method (ANM) associated to Pad´e approximants. This technique is very efficient in solving nonlinear problems in particular for instability modeling of thin structures thanks to the high order algorithm leading to a high accuracy in computing singular points along the nonlinear solution branches. We present different techniques to detect bifurcation points. The first technique is based on a bifurcation indicator introduced in the nonlinear problem in the form of a scalar function representing the intensity of a fictitious perturbation force which is evaluated along the equilibrium branch and which vanishes exactly at singular points. The second technique is based on Pad´e approximants that can be used as a bifurcation indicator by analyzing the denominator of rational fractions. The bifurcation corresponds to the first real root of the denominator. A third technique consists in a combination of buckling and linear vibrations which allows the extraction of singular points by analyzing the evolution of natural frequencies along the equilibrium path. Several numerical examples show the efficiency and robustness of the proposed methods.