We prove that the set of \gamma-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all \gamma \neq 0. Our proof relies on the coupling between a GFF and the nested CLE_4. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE_4 nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE_{\kappa}, \kappa \in (8/3,4]. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.
