Majorana

Neutrinos are so hard to detect and so low mass that one question that you'd think was easy hasn't been answered in the 80 years since Majorana proposed it: does a neutrino have a distinct anti-particle? Electrons do, no question, likewise protons, muons, and so on. Photons don't. The "anti-particle" for a photon would be another photon exactly out of phase with it. (Which is to say that the photon would never exist at all, since it would be cancelled at every point by the other photon.) You could say similarly that a π+ and π- are each others' antiparticles; they are composite particles and their quark content is the mirror image of the other (up and anti-down quark vs anti-up and down).

But the neutrino is so hard to pin down that we can't tell. What would you get if two did annihilate? They're neutral, so photons don't couple to them very easily. (Photons can couple to "W" particles which in turn couple to neutrinos, but adding more links in the chain makes it less likely to happen.)

You can try a trick.

But first...

When a nucleus has a radioactive decay that releases an electron (or a positron which is just an anti-electron), it also releases a neutrino. Since momentum and energy are conserved, the recoil of the nucleus and the energy of the electron tell you about the energy of the neutrino (which... ooops... you didn't see, so we call it "missing energy" Creative names, right?). Put the atom in a crystal lattice so it can't recoil easily (so its kinetic energy will be trivially small). That makes the problem much easier; it is more like a 2-body problem. Now measure the energy of the electron. Do it again and again and again and look at the distribution. If the neutrino's mass were a little larger you could see the edge of the distribution of electron energies drop to zero a little faster than if it were massless. That's a tough experiment, and hasn't been successful yet. We know the neutrino has mass, but it is so small that this method hasn't spotted it yet.

Now pick something even rarer: nuclear decays that toss out 2 electrons at once (not one after another!) Only a handful of nuclei do that. Most of the time you expect 2 neutrinos as well, but if the neutrino is its own antiparticle sometimes the neutrinos will annihilate each other and you only get 2 electrons, with no "missing energy". These experiments have been tried for years without success, but there's a new one that's a little more ambitious. It is still a prototype, but the final system (if it works well enough to build) will have a ton of germanium-76 crystals inside multiple layers of shielding and detectors. Natural radioactivity in the rock will add a background of random energy in the system unless you shield against it carefully, and cosmic rays dump in energy and sometimes catalyze nuclear reactions to boot--so you want to detect them on the way in and veto against them: "A cosmic ray just went through the system so don't look at anything for a few microseconds."

If Majorana was right the Standard Model of particles will need some revision. We know the model isn't complete, but it works very well anyhow. Any revision has a hard row to hoe to do as well.