Japanese neutrino experiment has first three candidate electron antineutrinos

This experiment should neither prove nor disprove it. It measures a parameter of a phenomenon (neutrino oscillation) which is well-known and which is only weakly affected by the premise that neutrinos are Majoranas.

So, just as a quick overview, there are two ways within the Standard Model to give a particle mass, one is the Higgs mechanism where the particle couples with a field generating an effective Dirac mass; another is to sum up certain Feynman diagrams (loops and other particles and such) generating a Majorana mass. These Majorana mechanisms effectively require that the particle be a charge-neutral fermion, so bosons (force-carriers) are out, as are quarks, electrons, muons, and taus. That leaves the neutrinos as the only known particles which could get mass this way.

For a while, it wasn't even known if neutrinos had mass; we now know that they do, but it's (m) at a scale much smaller than any of the known masses (M) of the Standard Model, and just postulating, "hey, the diagrams work out just right so m/M is a small number" feels like a cop-out. So one of the most intriguing Majorana-mechanisms is the so-called "seesaw mechanism": this postulates some symmetry breaking between the left-handed neutrinos of the Standard Model and some new right-handed neutrinos; the left-handed get ratio m/M and the right-handed I think get ratio M/m and it all sort of balances out.

There are two clear "signatures" that neutrinos are Majoranas: one would be to observe these huge right-handed neutrinos. The other is to observe two neutrinos perfectly annihilate.

What I think you're thinking of: In the early 2000s there were claims that experiments had observed the latter. This is a little tricky and the process is called "neutrinoless double beta decay," since a beta-decay produces an antineutrino and two antineutrinos can only annihilate if they are their own antiparticle.

The general consensus now is that those early experiments weren't sufficiently reproducible and we've never seen neutrinoless double-beta decay. However, that doesn't shut the door completely since double-beta-decay is already pretty rare: 76-Ge for example decays via double-beta-decay into 76-Se with a half-life of 10^21 years, or a hundred billion times the age of the universe. The GERDA experiment has been studying this stuff and can conclusively say that the half-life for neutrinoless double beta decay must be at least 10^25 years, and they're looking to increase their resolution to 10^26 years: so it's less than 1/10,000 of their decays observed so far, but they can capture it if it's more frequent than 1/100,000.

Drostie's post provides an excellent summary of the process of measuring this, though there are more experiments than GERDA in the works - two that come to mind immediately are SuperNEMO (https://en.wikipedia.org/wiki/Neutrino_Ettore_Majorana_Obser...) and SNO+ (https://en.wikipedia.org/wiki/SNO%2B) - Additionally I'd be remiss in not mentioning COBRA (http://www.cobra-experiment.org/) which was around a few years ago but I'm not sure of the current status.

As far as the "Popularity" - I'm sure there are groups of theoretical physicists that work purely on the assumption of one or the other, but most treatments I've seen cover both the cases for Dirac and Majorana neutrinos, given that we don't have evidence one way or the other.