Since we now know that neutrinos have a rest-mass, we ought to be able to observe relatively slow-moving neutrinos. Have we seen any?
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The energy of neutrinos depends on the process that formed them (more energetic reactions create more energetic neutrinos). Since neutrinos have no charge, there is no way to use electric fields to accelerate them (we can do that with charged particles in colliders).
We may see neutrinos of the Cosmic neutrino background (CNB), which have so little energy to be non-relativistic (for a comparison of neutrino energies and their cross-section according to the process that created them, see this plot).
There are real proposals for detectors of neutrinos from the CNB, but we have detected none to date (as far as I know).
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They haven't been seen and I doubt they will ever be. Since they were created they have traveled almost undisturbed. I find it difficult to believe most neutrinos from the Sun travel through the Earth. But it seems to be the case. Because they have mass and interact (though weak), they should be stoppable. Maybe in between the space between the two stars of a binary system, they could collide head on inelastically. If the stars emit the same number of neutrinos as the Sun, this should happen once in a while, though I'm not sure if this can happen inelastically.
We're not sure though. Of the zillions there might actually be some slow ones.
