We consider a family of random dynamical systems, each consisting of two piecewise affine increasing homeomorphisms $f_-, f_+$ of the unit interval, each with exactly one point of non-differentiability, iterated randomly according to given probability vector. Since systems of this type were considered by Alsedà and Misiurewicz, we call them Alsedà-Misiurewicz systems, or AM-systems. Under certain assumptions, such a system admits a unique stationary probability measure $\mu$ with no atoms at the endpoints. In this case, $\mu$ has to be either singular or absolutely continuous with respect to the Lebesgue measure. We prove singularity and calculate Hausdorff dimension of the measure $\mu$ and its support for systems satisfying some resonance conditions. We also prove that $\mu$ is singular for a certain open set of parameters, verifying a conjecture by Alsedà and Misiurewicz in this case. This is joint work with Krzysztof Barański.