The dynamics of spherical vapor bubbles in a standing acoustic wave are considered using the second order approximation with respect to the amplitude of the acoustic field. Qualitative analysis of the stability of vapor bubbles is performed, and computations of bubble dynamics in normal and reduced gravity conditions are carried out. It is found that vapor bubbles in standing waves are unstable in saturated and superheated water at 100°C, but potentially can be stabilized in subcooled liquids. Such stabilization can occur in a narrow range of parameters including reduced gravity conditions. Stable low frequency oscillations of the average bubble radius and position are also predicted under certain conditions.