  We have observed the remnant of supernova SN~1987A (SNR~1987A), located in
the Large Magellanic Cloud (LMC), to search for periodic and/or transient radio
emission with the Parkes 64\,m-diameter radio telescope. We found no evidence
of a radio pulsar in our periodicity search and derived 8$\sigma$ upper bounds
on the flux density of any such source of $31\,\mu$Jy at 1.4~GHz and
$21\,\mu$Jy at 3~GHz. Four candidate transient events were detected with
greater than $7\sigma$ significance, with dispersion measures (DMs) in the
range 150 to 840\,cm$^{-3}\,$pc. For two of them, we found a second pulse at
slightly lower significance. However, we cannot at present conclude that any of
these are associated with a pulsar in SNR~1987A. As a check on the system, we
also observed PSR~B0540$-$69, a young pulsar which also lies in the LMC. We
found eight giant pulses at the DM of this pulsar. We discuss the implications
of these results for models of the supernova remnant, neutron star formation
and pulsar evolution.
