  Eccentric Keplerian discs are believed to be unstable to three-dimensional
hydrodynamical instabilities driven by the time-dependence of fluid properties
around an orbit. These instabilities could lead to small-scale turbulence, and
ultimately modify the global disc properties. We use a local model of an
eccentric disc, derived in a companion paper, to compute the nonlinear vertical
("breathing mode") oscillations of the disc. We then analyse their linear
stability to locally axisymmetric disturbances for any disc eccentricity and
eccentricity gradient using a numerical Floquet method. In the limit of small
departures from a circular reference orbit, the instability of an isothermal
disc is explained analytically. We also study analytically the small-scale
instability of an eccentric neutrally stratified polytropic disc with any
polytropic index using a WKB approximation. We find that eccentric discs are
generically unstable to the parametric excitation of small-scale inertial
waves. The nonlinear evolution of these instabilities should be studied in
numerical simulations, where we expect them to lead to a decay of the disc
eccentricity and eccentricity gradient as well as to induce additional
transport and mixing. Our results highlight that it is essential to consider
the three-dimensional structure of eccentric discs, and their resulting
vertical oscillatory flows, in order to correctly capture their evolution.
