Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on $X_0(pM)$ (for $M$ such that $X_0(M)$ has genus zero and $p$ prime with $p\nmid M$), and he proved that if $Q$ is a $\mathbb{Q}$-rational Weierstrass point on $X_0(pM)$, then its reduction modulo $p$ is supersingular. The paper shows that, for square-free $M$ on the list, all supersingular $j$-invariants are obtained in this way. Furthermore, for most cases where $M$ is prime, the explicit correspondence between Weierstrass points and supersingular $j$-invariants in characteristic $p$ is described. Along the way, a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo $p$ is generalized.