The wave equation □ g M , a ψ = 0 on subextremal Kerr spacetimes ( M M , a , g M , a ) , 0 < | a | < M , does not admit real mode solutions, as was established by Shlapentokh-Rothman. In this paper, we show that the absence of real modes does not persist under the addition of an arbitrary short-range non-negative potential V to the wave equation or under changes of the metric g M , a in the far away region of M M , a (retaining the causality of the stationary Killing field T there). In particular, we first establish, for any 0 < | a | < M , the existence of real mode solutions ψ to equation □ g M , a ψ − V ψ = 0 , for a suitably chosen time-independent real potential V with compact support in space, satisfying V ≥ 0 . Exponentially growing modes are also obtained after perturbing the potential V . Then, as an application of the above result, we construct a family of spacetimes ( M M , a , g M , a ( d e f ) ) which are compact in space perturbations of ( M M , a , g M , a ) , have the same symmetries as ( M M , a , g M , a ) and moreover admit real and exponentially growing modes. These spacetimes contain stably trapped null geodesics, but we also construct a more complicated family of spacetimes with normally hyperbolic trapped set, admitting real and exponentially growing modes, at the expense of having conic asymptotics. The aforementioned results are in contrast with the case of stationary asymptotically flat (or conic) spacetimes ( M , g ) with a globally timelike Killing field T , where real modes for equation □ g ψ − V ψ = 0 are always absent, giving a useful zero-frequency continuity criterion for showing stability for a smooth family of equations □ g ψ − V λ ψ = 0 , with λ ∈ [ 0 , 1 ] and V 0 = 0 . We show explicitly that this criterion fails on Kerr spacetime. [ABSTRACT FROM AUTHOR]