We have searched for microsecond-timescale broadband emission from a sample of 18 millisecond pulsars. Our study places strong limits on such emission from several millisecond pulsars and shows that it is only present in a small subset of millisecond pulsars. Giant pulses of up to 64 times the mean pulse energy were detected from PSR J1823-3021A in the globular cluster NGC 6624. In contrast to the giant pulses of PSR B1937+21, nearly all of the giant pulses from PSR J1823-3021A were distributed within the trailing half of the main-pulse component of the integrated pulse profile. The fact that no giant pulses were observed on the leading side of the main-pulse component suggests that giant pulses are preferentially emitted closer to the last open field line than ordinary emission. The correlation between giant pulse emissivity and spin-down luminosity in millisecond pulsars suggests that the high period derivative of PSR J1823-3021A is intrinsic and is not just an artifact of its acceleration in the gravitational potential of NGC 6624.