We perform, by utilizing spectroscopic measurements made with SOHO/UVCS, a morphological characterization of the quiescent streamer during solar minimum, by means of both a comprehensive survey of daily synoptic observations over 33 weeks from 1996 April to November and detailed analysis of several specialized observations targeting quiescent streamers. On more than half of the days in our synoptic survey, we detected the O VI bifurcation, a feature for which the O VI λ103.2 streamer image has a marked contrast with that of H I Lyα, with the former showing intensity reduction along the streamer axis. All quiescent streamers observed in our synoptic survey exhibited the O VI bifurcation, and all streamers that did not exhibit the bifurcation were associated with active regions. Thus, it can be inferred that the O VI bifurcation is likely a fundamental physical characteristic of quiescent streamers. This suggests that its occurrence represents a basic aspect of the equatorial corona during solar minimum and therefore it can be regarded as an intrinsic nature of the interplanetary neutral sheet on the border of its solar side. Within the quiescent streamer the region of larger H I Lyα widths is found to be organized in an inverted heart-shaped configuration. At the lateral side of the legs and, for larger heights (>2.1 R☉), even within the core, spectral profiles of the O VI λλ103.2, 103.7 doublet have broad wings and thus cannot be fitted well with a single Gaussian, suggesting the presence of the broad component at those locations.