The analytic formulation of optimal stopping problems naturally leads to the study of variational inequalities. However, the existence of classical solutions to these equations is typically precluded: depending on the properties of the underlying diffusion, the value function may fail to attain $C^2$ or even $C^1$ regularity across the free boundary. To investigate the regularity of the value function, this seminar employs the framework of mild solutions. We introduce a penalization scheme, establishing the existence of approximating solutions and their convergence to the true value function. The regularity of these approximants is subsequently derived by exploiting the smoothing properties of the transition semigroup. Finally, under suitable admissibility assumptions on the obstacle, we show how this regularity is transferred from the approximating sequence directly to the value function.