We have gain-switched GaAs vertical-cavity surface-emitting lasers (VCSEL's) using sinusoidal electrical modulation at rates between 1.5 and 8 GHz, using devices with operating wavelengths between 820 and 860 nm. The shortest pulse obtained directly from such a laser was 24 ps. The time-bandwidth products for the gain-switched VCSEL's was between 0.6 and 3, which is smaller than the time-bandwidth products observed for gain-switched, single-frequency, edge-emitting lasers. Some of the excess bandwidth is caused by linear chirp, which was compensated using linear dispersion in single-mode optical fiber. The shortest compressed pulse was 15 ps. The pulses contained significant nonlinear chirp, however, which reduced the expected compression factor for linear dispersion to a factor of 2. The timing jitter for gain-switched pulse trains was 4-6 ps, which is comparable to the timing jitter observed for gain-switched, single-frequency, edge-emitting lasers. We discuss device design tradeoffs which affect the duration of the pulses from gain-switched VCSEL's. In particular, to minimize pulse duration, it is necessary to minimize the photon lifetime, which generally increases the threshold gain. Such an increase is deleterious because of the significant resistive heating in the DBR stacks of the monolithic device. Given the present resistivity of the DBR stacks, the design of the VCSEL's studied here is about optimal for short pulse generation.