  We investigate the boundedness of unimodular Fourier multipliers on
modulation spaces. Surprisingly, the multipliers with general symbol
$e^{i|\xi|^\alpha}$, where $\alpha\in[0, 2]$, are bounded on all modulation
spaces, but, in general, fail to be bounded on the usual $L^p$-spaces. As a
consequence, the phase-space concentration of the solutions to the free
Schr\"odinger and wave equations are preserved. As a byproduct, we also obtain
boundedness results on modulation spaces for singular multipliers
$|\xi|^{-\delta} \sin (|\xi|^\alpha)$ for $ 0 \leq \delta \leq \alpha $.
