We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation.