  We point out that dark-energy perturbations may become unstable in the
presence of a gravitational wave of sufficiently large amplitude. We study this
effect for the cubic Horndeski operator (braiding), proportional to
$\alpha_{\rm B}$. The scalar that describes dark-energy fluctuations features
ghost and/or gradient instabilities for gravitational-wave amplitudes that are
produced by typical binary systems. Taking into account the populations of
binary systems, we conclude that the instability is triggered in the whole
Universe for $|\alpha_{\rm B} |\gtrsim 10^{-2}$, i.e. when the modification of
gravity is sizeable. The instability is triggered by massive black-hole
binaries down to frequencies corresponding to $10^{10}$ km: the instability is
thus robust, unless new physics enters on even longer wavelengths. The fate of
the instability and the subsequent time-evolution of the system depend on the
UV completion, so that the theory may end up in a state very different from the
original one. The same kind of instability is present in beyond-Horndeski
theories for $|\alpha_{\rm H}| \gtrsim 10^{-20}$. In conclusion, the only
dark-energy theories with sizeable cosmological effects that avoid these
problems are $k$-essence models, with a possible conformal coupling with
matter.
