An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.