A colleague and I were discussing the fact that beta decay can emit neutrinos with arbitrarily low energies. We have neutrino detectors that can detect solar neutrinos -- and maybe extrasolar neutrinos or a cosmological component as well? -- but presumably any such detector has some energy threshold. E.g., if the working volume is made of chlorine, then is the detector only sensitive to neutrinos with enough energy to transmute 35Cl or 37Cl into 35- or 37- S or Ar? If so, then how far down in energy does our knowledge of the ambient neutrino environment go, and can we only guess about the spectrum below that based on extrapolation?
Rob's comment pointed me to some relevant information, so I thought I would write up a quick self-answer. Since this isn't my field, I may be getting stuff wrong here. A more authoritative answer from a specialist would be great, as would comments correcting any mistakes I may be making.
Borexino is the first neutrino detector to be able to measure neutrinos with energies below 3 MeV. Its energy range includes of the peak of the spectrum of solar neutrinos from the pp process. It consists of 278 tons of liquid organic scintillator, shielded by a large volume of water. Neutrinos scatter inelastically from electrons in the scintillator, so there is no minimum energy needed in order to interact. Below about 500 keV, it mostly sees background from 14C and 210Po.
For comparison, the detector Super-Kamiokande used both scattering from electrons and scattering from nuclei, which produced ultrarelativistic electrons and positrons. It was only sensitive to events that were energetic enough for the electron or proton to produce Cerenkov radiation.
Below is an energy spectrum:
Here are a couple of a recent papers on Borexino's results: