We study self-adjoint operators of the form $H_\omega=H_0{+}\sum \omega(n)(\delta_n|\cdot)\delta_n$ , where the $\delta_n$ 's are a family of orthonormal vectors and the $\omega(n)$ 's are independent random variables with absolutely continuous probability distributions. We prove a general structural theorem that provides in this setting a natural decomposition of the Hilbert space as a direct sum of mutually orthogonal closed subspaces, which are a.s. invariant under $H_\omega$ , and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of $H_\omega$ is a.s. simple