  Recently, there have been reports of six bright, dispersed bursts of coherent
radio emission found in pulsar surveys with the Parkes Multi-beam Receiver. Not
much is known about the progenitors of these bursts, but they are
highly-energetic, and probably of extragalactic origin. Their properties
suggest extreme environments and interesting physics, but in order to
understand and study these events, more examples need to be found. Fortunately,
the recent boom in radio astronomy means many 'next-generation' radio
telescopes are set to begin observing in the near future. In this paper we
discuss the prospects of detecting short extragalactic bursts, in both
beamformed and imaging data, using these instruments. We find that often the
volume of space probed by radio surveys of fast transients is limited by the
dispersion measure (DM) of the source, rather than its physical distance
(although the two quantities are related). This effect is larger for
low-frequency telescopes, where propagation effects are more prominent, but,
their larger fields-of-view are often enough to compensate for this. Our
simulations suggest that the low-frequency component of SKA1 could find an
extragalactic burst every hour. We also show that if the sensitivity of the
telescope is above a certain threshold, imaging surveys may prove more fruitful
than beamformed surveys in finding these sorts of transients.
