We report here an extension of a previous work in which we have shown that matrix models provide a tool to compute the intersection numbers of p-spin curves. We discuss further an extension to half-integer p, and in more details for $p=\frac{1}{2}$ and $p=\frac{3}{2}$. In those new cases one finds contributions from the Ramond sector, which were not present for positive integer p. The existence of Virasoro constraints, in particular a string equation, is considered also for half-integral spins. The contribution of the boundary of a Riemann surface, is investigated through a logarithmic matrix model. The supersymmetric random matrices provide extensions to mixed positive and negative p punctures.