  Strongly lensed supernovae can be detected as multiply imaged or highly
magnified transients. In order to compare the performances of these two
observational strategies, we calculate expected discovery rates as a function
of survey depth in five grizy filters and for different classes of supernovae
(Ia, IIP, IIL, Ibc and IIn). We find that detections via magnification is the
only effective strategy for relatively shallow pre-LSST surveys. For survey
depths about the LSST capacity, both strategies yield comparable numbers of
lensed supernovae. Supernova samples from the two methods are to a large extent
independent and combining them increases detection rates by about 50 per cent.
While the number of lensed supernovae detectable via magnification saturates at
the limiting magnitudes of LSST, detection rates of multiply imaged supernova
still go up drastically at increasing survey depth. Comparing potential
discovery spaces, we find that lensed supernovae found via image multiplicity
exhibit longer time delays and larger image separations making them more
suitable for cosmological constraints than their counterparts found via
magnification.
  We find that the ZTF will find about 2 type Ia and 4 core-collapse lensed
supernovae per year at a limiting magnitude of 20.6 in the r band. Applying a
hybrid method which combines searching for highly magnified or multiply imaged
transients, we find that LSST will detect 89 type Ia and 254 core-collapse
lensed supernovae per year. In all cases, lensed core-collapsed supernovae will
be dominated by type IIn supernovae contributing to 80 per cent of the total
counts, although this prediction relies quite strongly on the adopted spectral
templates for this class of supernovae. Revisiting the case of the lensed
supernova iPTF16geu, we find that it is consistent within the 2\sigma contours
of predicted redshifts and magnifications for the iPTF survey.
