We investigate electron-positron pair production from vacuum for short laser pulses with a subcycle structure, in the nonperturbative regime (Schwinger pair production). We use the nonequilibrium quantum kinetic approach and show that the momentum spectrum of the created electron-positron pairs is extremely sensitive to the subcycle dynamics-depending on the laser frequency omega, the pulse length tau, and the carrier phase varphi-and shows several distinctive new signatures. This observation could not only help in the design of laser pulses to optimize the experimental signature of Schwinger pair production but also ultimately lead to new probes of light pulses at extremely short time scales.