The aim of my talk is to present the work of my PhD Thesis and my current research. It is concerned with local/global bifurcation of standing wave solutions to some nonlinear Schr\"odinger equations in $\mathbb{R}^N \ (N\geq1)$ and with stability properties of these solutions. The equations considered have a nonlinearity of the form $V(x)|\psi|^{p-1}\psi$, where $V:\mathbb{R}^N\to\mathbb{R}$ decays at infinity and is subject to various assumptions. In particular, $V$ could be singular at the origin.
Local/global smooth branches of solutions are obtained for the stationary equation by combining variational techniques and the implicit function theorem. The orbital stability of the corresponding standing waves is studied by means of the abstract theory of Grillakis, Shatah and Strauss.