  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.
