In this work, the spectral properties of a singly charged vortex in a Bose–Einstein condensate confined in a highly anisotropic (disc-shaped) harmonic trap are investigated. Special emphasis is placed on the analysis of the so-called anomalous (negative energy) mode of the Bogoliubov spectrum. We use analytical and numerical techniques to illustrate the connection of the anomalous mode to the precession dynamics of the vortex in the trap. Effects due to inhomogeneous interatomic interactions and dissipative perturbations motivated by finite-temperature considerations are explored. We find that both of these effects may give rise to oscillatory instabilities of the vortex, which are suitably diagnosed through the perturbation-induced evolution of the anomalous mode, and monitored by direct numerical simulations.