The temporal evolution of microwave pulses transmitted through random dielectric samples is obtained from the Fourier transform of field spectra. Large fluctuations are found in the local or single-channel delay time, which is the first temporal moment of the transmitted pulse at a point in the output speckle pattern. Both positive and negative values of the local delay time are observed. The widest distribution is found at low-intensity values near a phase singularity in the transmitted speckle pattern. In the limit of long duration, narrow-bandwidth incident pulses, the single-channel delay time equals the spectral derivative of the phase of the transmitted field.