The hydrostatic Euler equations are derived from the incompressible Euler equations by means of the hydrostatic approximation. Among the different stability criteria that arise in the study of linear stability for the incompressible Euler equations, we can mention Rayleigh’s stability criterion, which gives rise to the local Rayleigh condition. Linear and nonlinear instability of the hydrostatic Euler equations around certain shear flows is well-known, as well as the finite time blow-up of certain solutions in the absence of the local Rayleigh condition. On the other hand, local existence, uniqueness and stability have been established in Sobolev spaces under the local Rayleigh condition. In this talk I will present new features of the solution to the hydrostatic Euler equations under the local Rayleigh condition; under certain assumptions, we establish either the breakdown of the local Rayleigh condition or the formation of singularities. Additionally, we get necessary conditions for global solvability in Sobolev spaces. As a byproduct, we show the independence of stationary solutions. Our proof relies on new monotonicity identities for the solution to the hydrostatic Euler equations under the local Rayleigh condition.