We have performed a pulsar search of 18 known or probable supernova remnants with the Arecibo 305 m radio telescope at 430 and 1420 MHz. The remnants were selected to have angular sizes less than or of order a few telescope beam diameters (∼10' at 430 MHz and ∼3' at 1420 MHz). In some cases, larger remnants with centrally brightened cores of this scale were also included. The observations were sensitive to pulsars with periods ≳1 ms and flux densities as low as ∼0.2 mJy. The targets included two probable plerionic objects and several shell-type remnants with centrally brightened X-ray profiles. No new pulsars were discovered in the survey, and we discuss some of the selection effects that make the interpretation of pulsar surveys of supernova remnants problematic. Effects of high pulsar velocities, uncertain distances, and high remnant brightness temperatures make it difficult to exclude the presence of a pulsar in most remnants.