The mathematical program with switching constraints (MPSC), which is recently introduced, is a difficult class of optimization problems since standard constraint qualifications are very likely to fail at local minimizers. MPSC arises from the discretization of optimal control problems with switching constraints which appears frequently in the field of control. Due to the failure of standard constraint qualifications, it is reasonable to propose some constraint qualifications for local minimizers to satisfy some stationarity conditions that are generally weaker than Karush-Kuhn-Tucker stationarity such as Mordukhovich (M-) stationarity. First we propose the weakest constraint qualification for M-stationarity of MPSC to hold at local minimizers. Then we extend some weak verifiable constraint qualifications for nonlinear programming to allow the existence of switching constraints, which are all strictly weaker than MPSC linear independence constraint qualification and/or MPSC Mangasarian-Fromovitz constraint qualification used in the literature. We show that these newly introduced constraint qualifications are sufficient for local minimizers to be M-stationary. Finally, the relations among MPSC tailored constraint qualifications are discussed.