Lorentz violation
Many theories of quantum gravity predict that spacetime has a fundamental structure at the Planck scale. While direct probes of the Planck scale are not possible, this fundamental structure could still have observable consequences. One of the most promising avenues of exploration has been the possibility that quantum gravity might violate Lorentz invariance, which is one of the fundamental symmetries of special relativity. If Lorentz violation is to be compatible with everyday experience, it must be strongly suppressed somehow. A natural method of suppression which has been extensively studied is to suppress Lorentz violation by some power of the energy E of a particle over the Planck energy EPlanck. Just as the Planck length is extremely small, the Planck energy is extremely large (1.22 x 1019 GeV). Therefore the ratio E/EPlanck is very small for everyday particles (or even those produced in accelerators) and Lorentz violation suppressed by a power of E/EPlanck is compatible with everyday experience. However, there are some ways that we can still measure the effect, as described below.
Astrophysical constraints
If Lorentz invariance is violated then astrophysical objects that accelerate particles to extremely high energies (such as the Crab nebula, pictured in the x-ray spectrum) are sensitive to Lorentz violation. My collaborators and I have used the observation of high energy synchrotron and inverse Compton radiation from the Crab to place strong limits on Lorentz violation, effectively ruling out Lorentz violation in effective field theories at order E/EPlanck (reference).
Other astrophysical phenomena that can be used to study possible Lorentz violation are gamma ray bursters (GRB's). Gamma rays from GRB's exhibit strong linear polarization. Certain types of Lorentz violation would destroy this polarization as the gamma rays travel from the GRB to us. Since the polarization still exists, this type of Lorentz violation is also strongly constrained (reference).
For list of other astrophysical phenomena that can be used to set limits on Lorentz violation, including the GZK cutoff and active galaxies, see what my collaborators and I affectionately call 'the long paper'.
Effective field theories
Whatever the fundamental structure of spacetime is, at energies much lower than the Planck scale it must reduce to the standard model plus small corrections. The standard model is an effective field theory, so an approach to Lorentz violation that does not draw on the fundamental nature of spacetime is to look for structures that give rise to Lorentz violation and can be incorporated into an effective field theory. I have studied theories with a preferred frame, which means that one class of observers moving through space is chosen to be 'special'. Such a frame can be incorporated into the framework of general relativity, as long as we satisfy diffeomorphism invariance (which is the fundamental symmetry of general relativity). The construction and consequences of such a theory can be read about here.
Hawking radiation
In the standard picture of Hawking radiation, wave modes that reach spatial infinity originate near the black hole event horizon and propagate outwards. As they travel outwards these modes experience exponential redshiftifting due to the gravitational field of the black hole. At late times, the modes that would be observed far from the hole had wavelengths shorter than the Planck length near the horizon. These trans-Planckian modes are incompatible with the belief that quantum gravity limits the smallest length of spacetime to be the Planck length. If, however, Lorentz invariance is violated then there is a resolution to this puzzle. The observed modes either come from inside the black hole (if the Lorentz violation makes the radiated particles superluminal) or modes come in from infinity, "bounce" near the horizon and propagate back out (the subluminal case). For a discussion of the effect of Lorentz violation on Hawking radiation see this paper.
Other interests
Fundamental structure of spacetime
The fundamental structure of space that might give rise to Lorentz violation is, of course, what one would ideally like to discover. There are many, many proposals - string theory, loop quantum gravity, causal sets, Regge calculus, etc. I have a strong interest in discrete models such as causal sets, loop quantum gravity, and Regge calculus that may give hints about how whatever the real structure is behaves.
Analog models of quantum field theory in curved space
In condensed matter, one can construct systems where the propagation of long wavelength phonons (sound waves) is very similar to the propagation of a scalar field in a curved Lorentzian spacetime. Such systems are called 'analog models'. It is even possible to construct analogies to black holes in this manner, where the phonons that travel past a certain point cannot return. For example, consider a fluid where long wavelength phonons in the fluid propagate with speed cs, which is analogous to the speed of light in these models. Now put this fluid in a pipe and change the shape of the pipe such that the speed v of the fluid is faster than cs in one section and slower in an adjacent section. A phonon can travel "back against the current" only up to a certain point, where the the fluid speed equals cs. After that the fluid flow carries it down the pipe. This point in the pipe therefore mimics a black hole event horizon, from which nothing can escape. Other black hole features such as Hawking radiation are also present in these models. Since these models give an example of a system that has a fundamental structure at very short distances (where the fluid description breaks down), yet has a pseudo-Lorentz invariance at long distances.