  Strong gravitational lensing of supernovae is exceedingly rare. To date, only
a handful of lensed supernovae are known. Despite this, lensed supernovae have
emerged as a promising method for measuring the current expansion rate of the
Universe and breaking the Hubble tension. We present an extensive search for
gravitationally lensed supernovae within the Zwicky Transient Facility (ZTF)
public survey, covering 15,215 transients with good light curves discovered
during four years of observations. We crossmatch a catalogue of known and
candidate lens galaxies with our sample and find three coincident sources,
which were due to chance alignment. To search for supernovae magnified by
unknown lenses, we test multiple methods suggested in the literature for the
first time on real data. This includes selecting objects with extremely red
colours, those that appear inconsistent with the host galaxy redshift, and
those with bright absolute magnitudes inferred from the host galaxy redshift.
We find a few hundred candidates, most of which are due to contamination from
activate galactic nuclei, bogus detections, or unlensed supernovae. The false
positive rate from these methods presents significant challenges for future
surveys. In total, 132 unique transients were identified across all of our
selection methods that required detailed manual rejection, which would be
infeasible for larger samples. Overall, we do not find any compelling
candidates for lensed supernovae, which is broadly consistent with previous
estimates for the rate of lensed supernovae from the ZTF public survey alone
and the number expected to pass the selection cuts we apply.
