We have carried out the largest and most unbiased search for hypercompact (HC) HII regions. Our method combines four interferometric radio continuum surveys (THOR, CORNISH, MAGPIS, and White2005) with far-infrared and sub-mm Galactic Plane surveys to identify embedded HII regions with positive spectral indices; 120 positive spectrum HII regions have been identified from a total sample of 534 positive spectral index radio sources. None of these HII regions, including the known HC HII regions recovered in our search, fulfills the canonical definition of an HC HII region at 5GHz. We suggest that the current canonical definition of HC HII regions is not accurate and should be revised to include a hierarchical structure of ionized gas that results in an extended morphology at 5GHz. Correlating our search with known ultracompact (UC) HII region surveys, we find that roughly half of detected UC HII regions have positive spectral indices, instead of more commonly assumed flat and optically thin spectra. This implies a mix of optically thin and thick emission and has important implications for previous analyses which have so far assumed optically thin emission for these objects. Positive spectrum HII regions are statistically more luminous and possess higher Lyman continuum fluxes than HII regions with flat or negative indices. Positive spectrum HII regions are thus more likely to be associated with more luminous and massive stars. No differences are found in clump mass, linear diameter, or luminosity-to-mass ratio between positive spectrum and non-positive spectrum HII regions.