We have made sensitive, high-resolution images of the fields of seven pulsar-powered nebulae with the VLA at 20 and 6 cm. Our goal was to resolve out the extended emission from the nebula and to look for any remaining point sources that could be candidates for a central pulsar powering the nebular emission. Several point sources were found but no compelling pulsar candidates were identified on the basis of their integrated properties (i.e., polarization, spectral index, scintillation, and location). In the remnant 3C 58 we report the discovery of an elongated ''wisp'' near the vicinity of a compact X-ray source. We argue that this feature is a termination shock from the relativistic wind of an unseen pulsar. The resulting images, with no clear detections, constitute some of the most sensitive searches made toward pulsar-powered nebulae to date. The flux density limits from this work are used, along with data from the literature on the known pulsar-supernova associations, to make a direct determination of the beaming fraction for young pulsars. We estimate that the beaming fraction of young pulsars in these nebulae is 61% +/- 13%. This result is intermediate between other independent determinations of the pulsar beaming fraction. We explore possible sources of error in our analysis and conclude that in order to have a beaming fraction of 100%, as preferred by some authors, the mean luminosity at 1.4 GHz of the undetected pulsars in our sample would have to be less than 1.0 mJy kpc2, significantly below current estimates for the luminosity of young pulsars.