We consider several classes of random self-adjoint and unitary operators and investigate their microscopic eigenvalue distribution. We show that some of these operators exhibit a transition in their microscopic eigenvalue distribution, depending on the properties of the corresponding spectral measures. In the case of pure point spectral measures, the microscopic eigenvalue distribution is Poisson (no correlation). As the spectral measures approach an absolutely continuous measure, the repulsion between the eigenvalues increases and the microscopic eigenvalue distribution converges to the clock (or "picket fence") distribution.