  In this work, we focus on the multiplicity of singular spectrum for operators
of the form $A^\omega=A+\sum_{n}\omega_n C_n$ on a separable Hilbert space
$\mathcal{H}$, for a self-adjoint operator $A$ and a countable collection
$\{C_n\}_{n}$ of non-negative finite rank operators. When $\{\omega_n\}_n$ are
independent real random variables with absolutely continuous distributions, we
show that the multiplicity of singular spectrum is almost surely bounded above
by the maximum algebraic multiplicity of eigenvalues of
$\sqrt{C_n}(A^\omega-z)^{-1}\sqrt{C_n}$ for all $n$ and almost all
$(z,\omega)$. The result is optimal in the sense that there are operators where
the bound is achieved. Using this, we also provide effective bounds on
multiplicity of singular spectrum for some special cases.
