Can neutrino detectors tell what direction the neutrinos came from?
Depends on the detection technology.
Yes
- Cerenkov based detectors (SNO and Super-Kamiokande for instance, as well a many cosmic ray neutrino detector) are direction sensitive, and this is one of the design considerations that drive the use of this tricky technique. The best results come from quasi-elastic reactions like $\nu_l + n \to l^- + p$.
- The calorimeter stacks used for beam based work have pretty good direction resolving power (this includes OPERA, if that is what prompted this question BTW). Many reaction contribute, but again charged-current quasi-elastic gives the best direction data.
- Liquid argon time projection chambers are a fairly new technology for which we have no full scale experiments, but test-beds have been deployed and the results are very good.
Sorta
In principle liquid scintillator detectors still get a Cernekov ring on quasi-elastic events, but in practice they are too washed out by the scintillation signal and especially by absorption/re-emission to be of much use
MiniBoonNe made a valiant effort to get some use from this, but the results were disappointing; most LOS detectors don't even try (KamLAND for instance--where to my knowledge no one has even tried). I think I did see a plot showing that the original Chooz experiment could tell the reactor side of the experiment from the off side, but they needed a large portion of their data set to do it so it would have been of minimal help on a per event basis.
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).
The really big detectors like IceCube have some chance to get Cerenkov imaging, but often get better results from time of flight data from high energy events.
No
- Radio-chemical methods (as in Ray Davis' Homestake Mine experiment) have no direction sensitivity even in principle.
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 event 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.