Is PMNS neutrino mixing allowed in the standard model? Some people say that the standard model does not allow for non-zero neutrino masses: it prevents renormalization, they say.
Other people say that massive neutrinos with PMNS mixing are allowed in the standard model.
Who is right?
 A: The latter bunch is right, but you have to shake both groups and demand to know what they mean by the SM, a  terminology famously "invented expressly to be systematically abused"!
If you take the SM to mean the standard chiral gauge theory SSBroken by a Higgs field, there is nothing preventing Dirac neutrino masses; historically, Weinberg in 1967 did not put them in, as they were not apparent experimentally, and you don't spatchcock a simple model including speculative states and couplings for no reason. Since the originalist conceit may sow confusion, early on after the advent of the Weinberg-Salam model, well-meaning adults (Pais and Treiman) introduced the adaptable and inclusive term SM to precisely avoid arid doctrinaire discussions on proprietary models. It is adapted to cover the (evolving!) contemporary understanding of the landscape, consistent with the basic telltale chiral gauge group structure...
Consequently, the theory has conceptual "slots" for them, and is quite comfortable with them. The PMNS mixing, then, including extensions for Majorana masses,  can be soundly a piece of the SM.
Admittedly, neutrino masses of meVs, so, 14 orders of magnitude away from  that of the top quark, although speculated to arise from similar couplings, are giving pause to students of renormalization in the SM; these have major problems with the stability of huge/minuscule ratio numbers under renormalization. But their methodological and conceptual problems ("naturalness") are theirs alone, and their starting points are not a dispositive part of the SM, a basically effective theory describing TeV-and-below physics.
