Second-order cone programming problems are a tractable subclass of convex optimization problems and there are known polynomial algorithms for solving them. Stochastic second-order cone programming problems have also been studied in the past decade and efficient algorithms for solving them exist. A new class of interest to optimization community and practitioners is the mixed-integer version of these problems that requires some variables are restricted to be integer. In this paper, we describe five applications leading to stochastic mixed-integer second-order cone programming problems. We also describe solution algorithms for solving stochastic mixed-integer second-order cone programming using cuts and relaxations by combining existing algorithms for stochastic second-order cone programming with extensions of mixed-integer second-order cone programming. The applications, which are the focus of this paper, include facility location, portfolio optimization, uncapacitated inventory, battery swapping stations, and berth allocation planning. Since mixed-integer programs are generally known to be NP-hard, bringing applications to the surface can detect tractable special cases and inspire for further algorithmic improvements in the future.