  We derive here a new stability criterion for two-fluid interfaces. This
criterion ensures the existence of "stable" local solutions that do no break
down too fast due to Kelvin-Helmholtz instabilities. It can be seen both as a
two-fluid generalization of the Rayleigh-Taylor criterion and as a nonlinear
version of the Kelvin stability condition. We show that gravity can control the
inertial effects of the shear up to frequencies that are high enough for the
surface tension to play a relevant role. This explains why surface tension is a
necessary condition for well-posedness while the (low frequency) main dynamics
of interfacial waves is unaffected by it. In order to derive a practical
version of this criterion, we work with a nondimensionalized version of the
equations and allow for the possibility of various asymptotic regimes, such as
the shallow water limit. This limit being singular, we have to derive a new
symbolic analysis of the Dirichlet-Neumann operator that includes an infinitely
smoothing "tail" accounting for the contribution of the bottom. We then
validate our criterion by comparison with experimental data in two important
settings: air-water interfaces and internal waves. The good agreement we
observe allows us to discuss the scenario of wave breaking and the behavior of
water-brine interfaces. We also show how to rigorously justify two-fluid
asymptotic models used for applications and how to relate some of their
properties to Kelvin-Helmholtz instabilities.
