  Majorana billiards are finitely sized, arbitrarily shaped superconducting
islands that host Majorana bound states. We study the fermion-parity switches
of the ground state of Majorana billiards. In particular, we study the density
and statistics of these fermion-parity switches as a function of applied
magnetic field and chemical potential. We derive formulae that specify how the
average density of fermion-parity switches depends on the geometrical shape the
billiard. Moreover, we show how oscillations around this average value is
determined by the classical periodic orbits of the billiard. Finally, we find
that the statistics of the spacings of these fermion-parity switches are
universal and are described by a random matrix ensemble, the choice of which
depends on the antiunitary symmetries of the system in its normal state. We
thus demonstrate that "one can hear (information about) the shape of a Majorana
billiard" by investigating its "fermion-parity switch spectrum".
