We study stationary clusters of vortices and antivortices in dilute pancake-shaped Bose-Einstein condensates confined in nonrotating harmonic traps. Previous theoretical results on the stability properties of these topologically nontrivial excited states are seemingly contradicting. We clarify this situation by a systematic stability analysis. The energetic and dynamic stability of the clusters is determined from the corresponding elementary excitation spectra obtained by solving the Bogoliubov equations. Furthermore, we study the temporal evolution of the dynamically unstable clusters. The stability of the clusters and the characteristics of their destabilizing modes only depend on the effective strength of the interactions between particles and the trap anisotropy. For certain values of these parameters, there exist several dynamical instabilities, but we show that there are also regions in which some of the clusters are dynamically stable. Moreover, we observe that the dynamical instability of the clusters does not always imply their structural instability, and that for some dynamically unstable states annihilation of the vortices is followed by their regeneration, and revival of the cluster.
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