The electrical characteristics of Schottky contacts on individual threading dislocations (TDs) with a screw-component in GaN substrates and the structures of these TDs were investigated to assess the effects of such defects on reverse leakage currents. Micrometer-scale platinum/GaN Schottky contacts were selectively fabricated on screw- and mixed-TD-related etch pits classified based on the pit size. Current-voltage (I-V) data acquired using conductive atomic force microscopy showed that very few of the screw TDs generated anomalously large reverse leakage currents. An analysis of the temperature dependence of the I-V characteristics established that the leakage current conduction mechanisms for the leaky screw TDs differed from those for the other screw and mixed TDs. Specifically, anomalous current leakage was generated by Poole-Frenkel emission and trap-assisted tunneling via distinctive trap states together with Fowler-Nordheim tunneling, with the mechanism changing according to variations in temperature and applied voltage. The leaky TDs were identified as Burgers vector b = 1c closed-core screw TDs having a helical morphology similar to that of other screw TDs generating small leakage currents. Based on the results, we proposed that the atomic-scale modification of the dislocation core structure related to interactions with point defects via dislocation climbing caused different leakage characteristics of the TDs.