This paper summarized the state of the evidence for one or more light massive sterile neutrinos in addition to the three massive "fertile" neutrinos that are known to exist.
The Theoretical Background Regarding Sterile Neutrinos
Precision electroweak experiments rule out the existence of more than three light "fertile" neutrinos that interact via the weak force.
But, any positive evidence, even if indirect, of beyond the Standard Model particles is always notable, and these data do not rule out particles that do not couple to either photons or weak force bosons (the W or Z bosons) called "sterile neutrinos" because of their lack of interaction with other particles of nature via any of the Standard Model forces.
This isn't a terribly radical idea, because even in the Standard Model, right handed particles don't interact via the weak force (and likewise, left handed antiparticles don't interact via the weak force). But, because quarks and charged leptons oscillate between left and right parity modes, and Standard Model neutrinos come only in parties that allow for weak force interactions, the lack of weak force interactions with right parity particles doesn't have many notable consequences until you get into the thick of quantum mechanical calculations.
Massive neutrinos naturally oscillate between different neutrino flavors in a manner that is a function of their respective masses and the probabilities set forth in the PMNS matrix (which can be parameterized into three mixing angles and one CP violation phase with Dirac neutrinos, plus two additional parameters in the case of Majorana mass neutrinos).
The mechanism of neutrino oscillation is not well understood. For example, it isn't clear if neutrino oscillation is mediated by W bosons or some other kind of boson, even though the relative masses and oscillation frequencies in a three neutrino model are increasingly well understood. So, it is possible given what we know now about the mechanism of neutrino oscillations that a neutrino that does not interact via the weak force could still oscillate back and forth with fertile neutrino flavors.
Data from cosmic background radiation surveys like WMAP and Planck have long pointed to the possibility of one (or, until recently) even two additional massive sterile neutrinos that oscillate as fourth or fifth neutrinos flavors with the three kinds known to the Standard Model.
To be clear, however, these light sterile neutrinos are not themselves dark matter candidates, even though heavier sterile neutrinos (particularly those with masses on the order of 2 keV, are attractive dark matter candidates. Light sterile neutrinos with masses on the order of 1 eV are too light and too "hot" to fill the role of dark matter in astronomy, although some theories propose three generations of sterile neutrinos, like the other Standard Model fermions, with one or two light sterile flavors enough to count as neutrinos for cosmology purposes and at least one other that serves as a dark matter candidate.
The Latest Evidence Tends To Disfavor A Reactor Anomaly Light Sterile Neutrino
On balance, however, the evidence for a light sterile neutrino flavor that oscillates back and forth with fertile neutrino flavors is getting weaker.
(1) The statistical strength of the reactor anomaly has been reduced to a statistically insignificant 1.4 sigma by the discovery that PMNS matrix neutrino mixing angle θ13 is higher than previously assumed.
The hints of sterile neutrinos include the “reactor anomaly”. This is the observation that the reactor ν¯e ﬂux measured by detectors that are only (10 – 100) m from reactor cores is ∼ 6% below the theoretically expected value. If the missing ﬂux has disappeared by oscillating into another ﬂavor or ﬂavors, this behavior, like that observed in LSND and MiniBooNE, points to a splitting ∆m2 larger than ∼ 0.1 eV2. A recent analysis ﬁnds that if one uses the now-known value of the mixing angle θ13, and takes into account ν¯e ﬂux measurements at detectors that are about 1 km from their reactors, the missing ﬂux is reduced to ∼4% of the theoretically expected value, and the signiﬁcance of the discrepancy is reduced to 1.4 σ. The story of the reactor anomaly no doubt will continue. It would be desirable to see if ν¯e ﬂux that is produced by a radioactive source, rather than a reactor, disappears as well.The PMNS matrix mixing angle θ13 was once widely assumed to be nearly zero, but is now known to be closer to 9 degrees. Generally speaking, discrepancies between experimentally data and Standard Model theoretical predictions of two sigma or more are ignored as random noise.
If there is a signal, and not just noise, however (which could be due to weak experimental power because neutrinos are so hard to detect, rather than because a sterile neutrino does not exist) at least one sterile neutrino mass flavor must have a mass that is at least the square root of 0.1 ev2, which is 0.316 eV. Thus, the lightest possible sterile neutrino flavor in a 3+1 neutrino flavor scenario that fits the reactor data with three fertile Standard Model neutrinos and one heavier sterile neutrino, would be at least five times as heavy as the heaviest of the three fertile neutrino mass eigenstates in a normal neutrino mass hierarchy. With respect to a mixing angle into a fourth neutrino flavor, that data are as follows:
The appearance probability . . . is reported to be 0.0026 by LSND. If the disappearance probability . . . is ∼ 0.06, as suggested by the reactor data, the constraint is very comfortably satisﬁed.(2) Non-Reactor Source Experiments Disfavor The Reactor Anomaly Sterile Neutrino Hypothesis
The ICARUS and OPERA experiments (among other things) look at neutrino sources other than nuclear reactors, which is useful because nuclear reactor neutrino sources are hard to model accurately because they are so complex. Their data disfavor a source for the reactor anomaly that is due to neutrino oscillations with a sterile neutrino flavor.
Some evidence that the low-energy νe excess may not be due to oscillation has come from the ICARUS and OPERA experiments. These experiments have searched for νµ → νe at L/E ∼ 35 m/MeV, an L/E larger than those of LSND and MiniBooNE, but such that one is still not very sensitive to oscillations driven by the squared-mass splittings among ν1,2,3, while being sensitive to those driven by splittings larger than ∼ 0.03 eV2 (i.e. 0.173 eV). From negative results, ICARUS and OPERA disfavor a νµ → νe origin of the low-energy νe excess reported by MiniBooNE, although the strength of this disfavoring has been questioned.UPDATE: OPERA constraints an a light sterile neutrino that oscillates with "active" neutrino flavors is found in figure 5 of this paper (which honestly is less clear than one might hope).
(3) The sum of neutrino masses with a sterile neutrino is ruled out by Plank data using model dependent assumptions.
Planck and WMAP cosmic background radiation data is mixed on this possibility. It is not inconsistent with there being four rather than three neutrino flavors, although three flavors are more likely than four given the data which produces a value for "Neff" that has a value about a quarter of a neutrino species in excess of the value predicted if there are exactly three neutrino species with a significant margin of error.
Five neutrino flavors, however, are strongly disfavored by the cosmic background radiation data, unless the heaviest sterile neutrino has a mass of more than about 10 eV. A sterile neutrino that heavy would be treated as dark matter rather than a neutrino for the purposes of cosmic background radiation cosmology models.
The most recent recent Plank data, however, excludes at roughly a 95% probability, a sum of the masses of all neutrino species combined of 0.23 eV or more, which rule out a sum of three neutrino mass species of 0.377 eV or more at a roughly 3.2 standard deviation level, which is a more than 99% probability that this hypothesis is not true. Yet, 0.377 eV is the minimum sum of the four neutrino masses with sum of masses of 0.06 eV which is the minimum sum of the three fertile neutrino masses, plus the minimum sterile neutrino mass that can fit reactor anomaly data.
Still, Kayser's paper notes that this measurement of the sum of all of neutrino species masses is based on model dependent assumptions about how neutrinos oscillate between flavors and mass states that fit all existing data (including simple 3+1 models) but might not be true if sterile neutrinos don't oscillate as naturally into fertile neutrinos as fertile neutrinos do into each other.
(4) A fifth neutrino species is more strongly disfavored.
As noted above, Planck and WMAP data strongly disfavor a fifth neutrino flavor (e.g. a 3+2 model with two sterile neutrino flavors that are heavier than the fertile neutrino flavors, or 1+3+1 model with one sterile neutrino that is lighter than the lightest fertile neutrino and one that is heavier than the heaviest fertile neutrino), keeping in mind that a sterile neutrino with mass in excess of about 10eV does not count for these purposes.
According to Kayser's paper, most of the reactor data can be fit almost as well with four neutrino species as with five.
On balance, it is more likely than not, given the evidence available right now, that all evidence in favor of a light sterile neutrino of the kind that would address the perceived reactor anomaly is really just experimental uncertainty and systemic error in prior experiments. But, the evidence is not so overwhelming that the possibility can be definitively overruled.
Numerous experiments are underway or planned, to get to the bottom of this question in the next few years to the next decade or so.
I personally expect that these experiments will reveal three fertile massive Dirac neutrinos with a normal mass hierarchy, significant CP violation, and no light sterile neutrinos. But, time will tell.
Off Topic Footnote: Pion Polarization Anomaly Was Probably Due To Experimental Error
What is a pion?
A pion is meson, which is a type of composite particle made out of a quark and an anti-quark. Charged pions have an up quark and an anti-down quark or visa versa. Neutral pions have a linear combination of up quarks and anti-up quarks, and down quarks and anti-down quarks. Pions are the lightest particles that include quarks, with masses of about 138 MeV for neutral pions and 140 MeV for charged pions, and are the longest lived hadrons (i.e. composite particles bound by gluons) other than protons and neutrons.
Charged pions, at least, can be polarized, since they are made up of electrically charged components that are not distributed homogeneously. Their polarization sheds light on the internal structure of mesons, thus confirming (or alternatively in tension with) the predictions of QCD (quantum chromodynamics aka strong force physics) regarding this internal structure.
New Pion Polarization Experimental Data Confirms Standard Model Predictions
In recent experimental physics news with a similar bottom line to recent reactor anomaly experimental data, a new high precision measurement of the polarization of pions at the COMPASS experiment (predominantly negatively charged pions) is consistent with the theoretical prediction derived from Standard Model QED and QCD calculations of this fairly simple system (background on the COMPASS experiment is available here).
If correct, this resolves tensions between the theoretically predicted result and the experimentally measured result in several previous, less precise experiments. Thus, another set of data points that supported possible beyond the Standard Model physics has once again been tentatively quashed by using more accurate experimental procedures.
FWIW, however, I'm not personally aware of any particular beyond the Standard Model theories that had been based upon, or relied upon, this anomaly, although the need for new physics to explain the old results which were in tension with the theoretical prediction is discussed, for example, here. Pion polarization is also relevant to parity conservation (or the lack thereof) in hadrons.