Stability is a fundamental issue in the study of astrophysical jets, which are believed to be magnetized. We carry out an extensive linear stability analysis of magnetized cylindrical jets in a global framework. We focus on characterizing the small-scale, internal instabilities
that are confined deep within the jet interior. Although large-scale jet integrity and coherence are desirable, some kind of internal instability is needed in order to explain several jet observations. We analyze the importance of the often overlooked thermal pressure gradient for
triggering instabilities in a region of the jet dominated by a toroidal magnetic field and a weak vertical field. Such regions are likely to occur far from the jet source and boundaries, and are potential sites of magnetic energy dissipation that is essential to explain the particle acceleration and radiation observed from astrophysical jets. We find that the eigenfunctions of the most unstable modes are radially localized, which allows us to propose a generic instability criterion that transcends the complex nature of magnetic field inside jets. A stronger, radially varying vertical field, however, complicates this criterion by providing additional stabilization against the thermal
pressure gradient. Nevertheless, we argue that the jet interiors generically should be subject to rapidly growing, small-scale instabilities, capable of producing current sheets that lead to dissipation. We also find instabilities that are sensitive to the background radial structure but have growth rates smaller than the localized modes.