Solar coronal plumes long seemed to possess a simple geometry supporting spatially coherent, stable outflow without significant fine structure. Recent high-resolution observations have challenged this picture by revealing numerous transient, small-scale, collimated outflows (jetlets) at the base of plumes. The dynamic filamentary structure of solar plumes above these outflows, and its relationship with the overall plume structure, have remained largely unexplored. We analyzed the statistics of continuously observed fine structure inside a single representative bright plume within a mid-latitude coronal hole during 2016 July 2-3. By applying advanced edge-enhancement and spatiotemporal analysis techniques to extended series of high-resolution images from the Solar Dynamics Observatorys Atmospheric Imaging Assembly, we determined that the plume was composed of numerous time-evolving filamentary substructures, referred to as plumelets in this paper, that accounted for most of the plume emission. The number of simultaneously identifiable plumelets was positively correlated with plume brightness, peaked in the fully formed plume, and remained saturated thereafter. The plumelets had transverse widths of 10 Mm and intermittently supported upwardly propagating periodic disturbances with phase speeds of 190-260 km/s and longitudinal wavelengths of 55-65 Mm. The characteristic frequency (3.5 mHz) is commensurate with that of solar p-modes. Oscillations in neighboring plumelets are uncorrelated, indicating that the waves could be driven by p-mode flows at spatial scales smaller than the plumelet separation. Multiple independent sources of outflow within a single coronal plume should impart significant fine structure to the solar wind that may be detectable by Parker Solar Probe and Solar Orbiter.