We determine the detailed thermodynamic behavior of vortices in the O(2) scalar model in two dimensions (2D) and of global monopoles in the O(3) model in 3D. We construct numerical techniques, based on cluster decomposition algorithms, to analyze the point defect configurations. We find that these criteria produce results for the Kosterlitz-Thouless temperature in agreement with a topological transition between a polarizable insulator and a conductor, at which free topological charges appear in the system. For global monopoles we find no pair unbinding transition. Instead a transition to a dense state where pairs are no longer distinguishable occurs at $T<T_c$, without leading to long-range disorder. We produce both extensive numerical evidence of this behavior as well as a semianalytic treatment of the partition function for defects. General expectations for N=D>3 are drawn, based on the observed behavior.