  We address a physically-meaningful extension of the Prandtl system, also
known as hyperbolic Prandtl equations. We show that the linearised model around
a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly
in time, we generate solutions that experience a dispersion relation of order
k^(1/3) in the frequencies of the tangential direction, akin the pioneering
result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was
of order k^(1/2)). We emphasise however that this growth rate does not imply
ill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects to
the original Prandtl equations in Gevrey-class m, with m > 2. By relaxing
certain assumptions on the shear flow and on the solutions, namely allowing for
unbounded flows in the vertical direction, we however determine a suitable
shear flow for the mentioned ill-posedness in Gevrey spaces.
