The hydrodynamic instability of isothermal, inviscid and axisymmetric accretion (and wind) flows with standing shock waves is studied against infinitesimal, isothermal and axisymmetric perturbations. The post-shock flow is assumed to be transonic so that accretion on to black holes can be treated. A global stability analysis shows that the value of an unperturbed quantity, nu = -(1/V)(dPHI(eff)/dr-c(s)2/r), at the post-shock side of the unperturbed shock location determines stability properties, where V, c(s) and PHI(eff) are, respectively, the radial flow velocity, the isothermal sound speed, and effective gravitational potential including the centrifugal force. If nu greater-than-or-equal-to 0 no unstable mode exists, in other words v < 0 is a necessary condition for axisymmetric instability. This criterion means that the flow may be unstable only when the fluid decelerates on the pre-shock side. It is found that the growth rates and frequencies of unstable modes are bounded by Absolute value of nu.