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The class of continuous-time linear Markovian jump singular systems with discontinuities has been dealt with. Based on the stochastic stability in mean square sense of the system under study, a sufficient condition is proposed for the design of an output feedback controller which guarantees that the closed-loop system is piecewise regular, impulse-free and stochastically stable in mean square sense. First, the proposed approach is established in terms of bilinear and linear matrix inequalities. Then, the sequential linear programming matrix method is used to convert the nonlinear and non-convex output feedback problem on a convex optimisation one. A numerical example is presented to show the usefulness of the proposed results.