We consider singular SDEs driven by symmetric stable processes and with measure valued drift in a Kato class. Since the drift may be a measure which is not absolutely continuous one needs to find a rigorous definition of solutions. This can, for instance, be achieved by employing an approximation scheme [Kim & Song, '14]. We show that we can equivalently reformulate the drift term in terms of the local time of the unknown process. To this end, we derive a  Tanaka-type formula for symmetric stable processes that are perturbed by an adapted, right-continuous processes of finite variation. Finally, we discuss weak and strong existence and uniqueness of solutions.