  We study the scalar, electromagnetic and gravitational perturbations of
planar AdS$_4$ black holes with NUT charge. In the context of the AdS/CFT
correspondence, these solutions describe a thermal quantum field theory
embedded in a G\"odel-type universe with closed time-like curves. For a given
temperature and NUT charge, two different planar Taub-NUT solutions exist, but
we show that only the one with a positive specific heat contributes to the
Euclidean saddle point in the path integral. By using the Newman-Penrose
formalism, we then derive the master equations satisfied by scalar,
electromagnetic and gravitational perturbations in this background, and show
that the corresponding equations are separable. Interestingly, the solutions
pile up in the form of Landau levels, and hence are characterized by a single
quantum number $q$. We determine the appropriate boundary conditions satisfied
by the master variables and using these we compute the quasinormal modes of
scalar and gravitational perturbations. On the other hand, electromagnetic
perturbations depend on a free parameter whose determination is problematic. We
find that all the scalar and gravitational QNM frequencies lie in the lower
half of the complex plane, indicating that these Taub-NUT spacetimes are
stable. We discuss the implications of these results in the light of the
AdS/CFT correspondence.
