  We study the stationary probability distribution of a system driven by shot
noise. We find that, both in the overdamped and underdamped regime, the
coordinate distribution displays power-law singularities in its central part.
For sufficiently low rate of noise pulses they correspond to distribution
peaks. We find the positions of the peaks and the corresponding exponents. In
the underdamped regime the peak positions are given by a geometric progression.
The energy distribution in this case also displays multiple peaks with
positions given by a geometric progression. Such structure is a signature of
the shot-noise induced fluctuations. The analytical results are in excellent
agreement with numerical simulations.
