The near-field distribution of a light wave in a metal-coated optical fiber probe under the illumination of femtosecond laser pulses is numerically simulated. By choosing the maximum of the calculated light field data and the time when the light field reaches the maximum in a time period of one light wave cycle at each spatial point, we obtain the amplitude and phase distributions of the light field near the probe. We find that the output amplitude distribution of y-polarization has roughly a circled-cross-like pattern with two arc-like zero-amplitude zones. The zeros of light intensity are the phase singularities at which the phase is undefined. By analyzing the output phase distribution of y-polarization, we obtain the temporal evolution of the phase singularities and their characteristics