On the basis of the quantum Sutton-Chen potential, the rapid solidification processes of liquid silver have been studied by molecular dynamics simulation for four cooling rates. By means of several analysis methods, the competitions and transitions between microstructures during the cooling processes have been analyzed intensively. It is found that there are two phase transitions in all simulation processes. The first one is from liquid state to metastable (transitional) body-centered cubic (bcc) phase. The initial crystallization temperature T-ic increases with the decrease of the cooling rate. The second one is from the transitional bee phase to the final solid phase. This study validates the Ostwald's step rule and provides evidence for the prediction that the metastable bee phase forms first from liquid. Further analyses reveal that the final solid at 273 K can be a mixture of hexagonal close-packed (hcp) and face-centered cubic (fee) microstructures with various proportions of the two, and the slower the cooling rate is, the higher proportion the fee structure occupies.