This monograph extends the notion of locally most powerful rank tests to non-regular cases. Through this no- tion one is led in a natural way to "non-standard" rank tests. A nearly complete analysis of the finite sample and asymptotic properties of such rank tests is presented. Also an adaptive test procedure is proposed and studied, and the results of a Monte Carlo simulation are given which provide strong evidence that it should perform well in many practical situations. An appendix derives the limit experiments needed to investigate the asymptotic optimality of these "non-standard" rank tests under local alternatives. The results in the appendix should also be of separate interest.