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 150-840 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 SNR1987A. As a check on the system, we also observed PSR B0540-69, a young pulsar that 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.