I was reading this question and got to thinking. Can neutrino detectors give us any clue where the neutrinos came from or when a supernova may occur? I was unsure and decided to ask that here.
Depends on the detection technology.
A colleague has shown in bench studies that with sufficient time resolution (on order of 0.1 ns) it is possible to resolve the Cernekov/scintillation ambiguity and to RICH in scintillator.
Of course, a sparely instrumented LOS detector won't ever have a chance, as was the case with Cowen and Rheins instrument and the non-proliferation monitors that people are experimenting with (no link 'cause I've only ever seen a colloquium and don't recall the name of the instrument).
Note that the direction sensitivity is always for the momentum of reaction products rather than the neutrino itself. In the case of high energy neutrinos the direction of the products can be highly correlated with that of the neutrino, but at lower energies this becomes less true and pointing information is increasingly only good in aggregate.
The experiments that participate in the Supernova Early Warning System are all direction sensitive in some degree or another as the plan is to both alert the light telescopes that an even may be coming and tell them what part of the sky to search.
Non-direction-sensitive detectors also attempt to monitor supernova neutrino pulses, but without direction sensitivity their data will be more useful during the postmortem analysis of the timing difference.
Disclosure: I was associated with KamLAND for several years and am currently associated with two LArTPC projects.
Yes - have a look at the University of Utah neutrino detector which uses paired tanks to detect the direction of a muon by observing the Čerenkov light response in each tank.