AbstractWe address the Prandtl equations and a physically meaningful extension known as hyperbolic Prandtl equations. For the extension, 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 $$\root 3 \of {k}$$ k 3 in the frequencies of the tangential direction, akin the pioneering result of Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) for Prandtl (where the dispersion was of order $$\sqrt{k}$$ k ). We emphasise, however, that this growth rate does not imply (a-priori) ill-posedness in Gevrey-class m, with $$m>3$$ m > 3 . We relate these aspects to the original Prandtl equations in Gevrey-class m, with $$m>2$$ m > 2 : We show that the result in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) determines a dispersion relation of order $$\sqrt{k}$$ k for a short time proportional to $$\ln (\sqrt{k})/\sqrt{k}$$ ln ( k ) / k . Therefore, the ill-posedness in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) in its generality is momentarily constrained to Sobolev spaces rather than extending to the Gevrey classes.