We present analysis of a homogeneous, optically selected, volume-limited (0.2<z<0.3) sample of 128 radio-quiet quasi-stellar objects (QSOs) recently observed at 6GHz with the Very Large Array (VLA) in A configuration (~0.33" resolution). We compare these new results to earlier (2010-2011) 6GHz observations with the VLA in C configuration (~3.5"). While all of these radio-quiet QSOs (RQQs) were unresolved on a 3.5" scale (~14kpc at z=0.25), we resolve notable complex subgalactic structures in about half of the RQQs at 0.33" resolution (~1.3kpc at z=0.25). By comparison of flux density measurements between the two sets of observations, we demonstrate that significant sub-galactic-scale radio structure is present in at least 70% of the RQQ population and that the central component accounts for an average of ~65% of the total detected radio power. One RQQ, J0935+4819, shows striking symmetric, double-lobed morphology and appears to be the first identified example of a radio-quiet QSO with FRII type morphology on ~arcsec scale (projected size of >~6kpc). In addition to revealing RQQ subgalactic morphology, we employ counterparts from legacy (FIRST at 1.4GHz) and recent (VLA Sky Survey at 3GHz) VLA surveys to investigate radio spectral indices and potential variability over decades-long timescales for a subset of the RQQs and for the cores of radio-intermediate and radio-loud sources in the parent sample of 178 QSOs. These results support the growing notion that the RQQ population is not a monolithic phenomenon but instead consists of a mixture of mainly starburst-powered and jet-powered galaxies.