  A general system of several ordinary differential equations coupled with a
reaction-diffusion equation in a bounded domain with zero-flux boundary
condition is studied in the context of pattern formation. These
initial-boundary value problems may have regular (i.e. sufficiently smooth)
stationary solutions. This class of {\it close-to-equilibrium} patterns
includes stationary solutions that emerge due to the Turing instability of a
spatially constant stationary solution. The main result of this work is
instability of all regular patterns. It suggests that stable stationary
solutions arising in models with non-diffusive components must be {\it
far-from-equilibrium} exhibiting singularities. Such discontinuous stationary
solutions have been considered in our parallel work [\textit{Stable
discontinuous stationary solutions to reaction-diffusion-ODE systems}, preprint
(2021)].
