The pair-production process in the presence of strong linearly polarized laser fields with a subcycle structure is considered. Laser pulses with different envelope shapes are examined by means of a nonperturbative numerical approach.As generic cases, we analyze two different "flat" envelope shapes and two shapes without a plateau as a function of various characteristic parameters including the carrier-envelope phase, respectively. The resonant Rabi oscillations, the momentum distribution of particles created, and the total number of pairs are studied. It is demonstrated that all these characteristics are very sensitive to the pulse shape.