The paper is concerned with the numerical solution of stochastic singular integral equations with random integral operators and with random functions in their right-hand sides. Two kinds of random solutions are studied here – in the broad sense and in the strict sense. In the second case the random solution is measurable. For a stochastic integral equation with the Cauchy kernel an explicit formula of all random solutions is found, and its unique solvability is proved. The numerical scheme proposed for solving stochastic integral equations is based on approximating the right-hand side and the regularized kernel by random polynomials. In doing that, the author represents the approximate solution as a generalized random interpolating polynomial with unknown coefficients, for which he obtains a system of linear stochastic algebraic equations. Such a scheme is also realized for a stochastic singular integral equation with Hilbert kernel. Some numerical experiments are shown.