  The excess of neutrino candidate events detected by IceCube from the
direction of TXS 0506+056 has generated a great deal of interest in blazars as
sources of high-energy neutrinos. In this study, we analyze the publicly
available portion of the IceCube dataset, performing searches for neutrino
point sources in spatial coincidence with the blazars and other active galactic
nuclei contained in the Fermi 3LAC and the Roma BZCAT catalogs, as well as in
spatial and temporal coincidence with flaring sources identified in the Fermi
Collaboration's All-Sky Variability Analysis (FAVA). We find no evidence that
blazars generate a significant flux of high-energy neutrinos, and conclude that
no more than 5-15% of the diffuse flux measured by IceCube can originate from
this class of objects. While we cannot rule out the possibility that TXS
0506+056 has at times generated significant neutrino emission, we find that
such behavior cannot be common among blazars, requiring TXS 0506+056 to be a
rather extreme outlier and not representative of the overall blazar population.
The bulk of the diffuse high-energy neutrino flux must instead be generated by
a significantly larger population of less-luminous sources, such as non-blazar
active galactic nuclei.
