  The impending era of wide-field radio surveys has the potential to
revolutionize our understanding of astrophysical transients. Here we evaluate
the prospects of a wide range of planned and hypothetical radio surveys using
the properties and volumetric rates of known and hypothetical classes of
extragalactic synchrotron radio transients (e.g., on- and off-axis gamma-ray
bursts [GRB], supernovae, tidal disruption events [TDE], compact object
mergers). Utilizing these sources and physically motivated considerations we
assess the allowed phase-space of radio luminosity and peak timescale for
extragalactic transients. We also include for the first time effects such as
redshift evolution of the rates, K-corrections, and non-Euclidean luminosity
distance, which affect the detection rates of the most sensitive surveys. The
number of detected events is calculated by means of a Monte Carlo method, using
the various survey properties (depth, cadence, area) and realistic detection
criteria that include a cut on the minimum variability of the transients during
the survey and an assessment of host galaxy contamination. Near-term GHz
frequency surveys (ASKAP/VAST, Very Large Array Sky Survey) will detect few
events: <~30-50 on- and off-axis long GRBs and off-axis tidal disruption
events, and ~10-20 neutron star binary mergers if ~1% of the mergers result in
a stable millisecond magnetar. Low-frequency surveys (e.g., LOFAR) are unlikely
to detect any transients, while a hypothetical large-scale mm survey may detect
~40 on-axis long GRBs. On the other hand, SKA surveys at ~0.1-1 GHz have the
potential to uncover thousands of transients, mainly on- and off-axis long
GRBs, on-axis short GRBs, off-axis TDEs, and neutron star binary mergers with
magnetar remnants.
