The Markov chain approximation method is a widely used numerical approach to computing optimal controls and value functions for general nonlinear jump diffusions, with a possible reflecting boundary. We extend the method to models with singular controls, where the control increment has the form g(x(t-))dH(t), which we call state dependent owing to the multiplier g(x). For the most part, past work concerned the case where g(.) is constant. There are major differences in the properties and treatments of the two cases. Owing to the possibility of "multiple simultaneous impulses," H(.) must be interpreted in a generalized sense, and the analysis must be done in a "stretched-out" time scale, analogously to the approach previously used by the author and colleagues.