We are interested in the question of stability in the field of shape optimization. We focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape functions, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations. These conditions are quite general and are satisfied by a lot of classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with volume constraint and translation invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local isoperimetric inequalities. We finally test the sharpness of our hypotheses by giving counterexamples of critical stable domains that are not local minima.