What measurements of neutrino mass have provided experimental deviations from the Standard Model?
Well, any measurement of mass for a neutrino is in contradiction with the standard model, as it assumed they were massless, hence moved at the speed of light.
The first confirmed detection was a collaboration between the Super-Kamiokande experiment in Japan and the Sudbury Neutrino Observatory in Canada in This won the discoverers the 2015 Nobel Prize in Physics.
This still leaves several problems, to say the least, for the Physics community. It didn’t find absolute masses for the neutrinos as I understand it, only the relationship (or difference) between the square of the masses. Suffice to say the masses are very, very low and this leaves open the possibility that neutrinos are Majorana particles (they are their own anti-particles.)
This article provides a brief introduction : https://phys.org/news/2015-12-nobel-winning-discovery-neutrino-oscillations-neutrinos.html
And this provides a much more in depth discussion (but the Maths is pretty horrible!): https://en.m.wikipedia.org/wiki/Neutrino_oscillation