It is probably easier to sum up the physics discoveries that almost surely won't be made in 2016 than it is to sum up those that are likely. Still, let's consider what active research is underway and what it is likely to find, or not.
* Foremost in the minds of particle physicists is the 750 GeV bump in both ATLAS and CMS searches found at the end of last year. There are already a mountain of pre-prints hypothesizing beyond the Standard Model explanations for it.
Now that the dust has settled, and there has been time to consider what it would take for this bump to be real, while no one has seen significant signs of it elsewhere, my personal estimation of the likelihood of it being real has fallen to about 35%.
For example, the extent to which the 125 GeV Higgs boson resembles the Standard Model Higgs boson in every respect, places strict bounds on the properties that a spin-0 or spin-2 electrically neutral boson with 750 GeV can exhibit consistent with existing data (the linked pre-print is also an exceptionally lucid introduction to the physics of the Higgs boson, generally before it goes on to speculate about the 750 GeV bump). The singlet boson model discussed in the linked paper would suggest a boson with couplings similar to the Higgs boson but at least 80% weaker. (UPDATE January 4, 2016: Jester estimates that it is at least 90% weaker. "From the non-observation of anything interesting in run-1 one can conclude that there must be little Higgsiness in the 750 GeV particle, less than 10%.").
The lack of a corresponding clear signal of a four-lepton channel of decay at 750 GeV is particularly notable. So is the lack of any channel with very large amounts of missing traverse energy at the LHC, which would be expected if there were decays to a dark sector or of stable particles such as the lightest supersymmetric particle.
These considerations are not insurmountable, but they force any beyond the Standard Model theory that can incorporate a 750 GeV particle to be particularly baroque. It is virtually impossible for the 750 GeV particle to be a new singlet particle, or for it to be an excitation of the Standard Model Higgs boson (which would have had four intermediate excitations before this one), consistent with the data to date. This means that if the 750 GeV bump is real, that there must be a whole suite of other new particles out there to be discovered along with it.
The analysis of the remaining data from 2015 and new data from 2016 should reveal if this bump grows in significance (as it should if it is real), or fades in significance (in which case it is almost surely a statistical fluke).
* Neutrino experiments will continue to refine the precision with which we know that parameters of neutrino oscillations and masses, but none of these will have enough data to be definitive on any question in 2016. We had a lot of the answers at the end of 2015 and will have to wait more than another year before we have more definitive answers in the area of neutrino physics.
* LIGO is rumored to have evidence of direct detection of gravitational waves, but I am highly skeptical of the report due to past false alarms and due to the theoretical expectation that gravity waves should be more subtle than what LIGO is capable of detecting.
* Astronomy is one area where there are myriad ongoing active experiments and a great deal of simulation work that could produce breakthroughs in dark matter/dark energy/inflation with dark matter in particular being susceptible to insights from new astronomy data. Black hole physics generally could also see progress.
Tommaso Dorigo points out another 750 GeV signal not seen: "the object does not decay to top pairs (otherwise we'd have seen a huge signal in X->tt searches)"
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