Summary: We are interested in the critical behavior of generalized Penner models at \(t\sim-1+\mu/N\), where the topological expansion for the free energy develops logarithmic singularities: \(\Gamma\sim-(\chi_0\mu^2\ln\mu+\chi_1\ln\mu+\cdots)\). We demonstrate that these criticalities can best be characterized by the fact that the large-\(N\) generating function becomes meromorphic with a single pole term of unit residue, \(F(z)\to 1/(z-a)\), where \(a\) is the location of the `sink'. For a one-band eigenvalue distribution, we identify multicritical potentials; we find that none of these can be associated with the \(c=1\) string compactified at an integral multiple of the self-dual radius. We also give an exact solution to the Gaussian Penner model and explicitly demonstrate that, at criticality, this solution does not correspond to a \(c=1\) string compactified at twice the self-dual radius.