The liquid jet penetration, breakup, and spray evaporation in crossflows, which find their importance in aeroegines, are investigated in the steady and oscillating air-crossflows using detailed numerical simulation to improve the understanding of the influence of the inflow oscillation that is commonly caused by the combustion instability on this complex process. To clarify the oscillation effects on jet atomization, the crossflow multimode breakup is considered, incorporating its unique characteristics of holding two different breakup behaviors, i.e., shear breakup and bag breakup. The oscillation frequency is set to be 500 Hz according to previous research on combustion instability. The results show that the liquid jet trajectory under steady air flow condition is in good agreement with the analytical solution and experimental correlation, around which the oscillating cases present a reasonable breakup behavior. The vortex morphology behind the liquid jet is found to largely affect the droplets evaporation and their downstream delivery. Moreover, the oscillating air inflow condition shows an enhanced atomization and evaporation property, producing the smaller Sauter mean diameter (SMD) and higher fuel vapor distribution at the leeward downstream region. This behavior of atomized spray in the oscillating crossflow suggests that the combustion instability could impose a significant effect on the subsequent spray combustion process.