There are exquisite laboratory tests of Lorentz invariance, for instance, by atomic physics experiments (see here, here, here, here, here, or here.) Nevertheless, there is widespread interest in the possibility that Lorentz symmetry is not an exact symmetry of nature. This interest arises partly because it is difficult to reconcile Lorentz invariance with the existence of a minimum length scale, an idea which some people consider necessary in the quantization of gravity. Some approaches to the quantization of gravity also seem to imply Lorentz violation; for instance, it has often been reported to be a natural consequence in Loop Quantum Gravity (see Lee Smolin's discussion or Google's listings about Loop Quantum Gravity).
I got interested in this problem, as a skeptic of the possibility of Lorentz violation, after discussions with Ann Nelson, who told me that constraints on Lorentz violation in gravitational physics are quite weak. We realized that much tighter constraints could be placed, using an idea proposed by Coleman and Glashow.
The idea of Coleman and Glashow is the following. Suppose that Lorentz symmetry is not a true symmetry of nature. This opens up the possibility that the limiting speeds (the highest speed which can be attained) of different particles, are different from each other. Suppose in particular that the speed of light (of electromagnetic radiation) and the limiting speed of a proton were not the same, and that the speed of a proton were higher. Then it turns out that the proton would lose energy to electromagnetic radiation, until its speed was the same as the speed of light.
The way this works is the following. Since the proton is electrically charged, it carries around an electromagnetic field. When the proton moves, the electromagnetic field of the proton must move with it. The emission of electromagnetic radiation (light) can be understood as that electromagnetic field continuing to propagate, when the proton's motion is changed by some external force; so every time a proton changes its speed or direction of motion, some of the electromagnetic field accompanying it continues in the old direction and becomes radiation. If a proton were moving faster than the speed of light, then it would "outrun" its own electromagnetic field; without any acceleration being necessary, the electromagnetic field of the proton would get stripped away from the proton and propagate away as photons (electromagnetic radiation). Since the proton is charged, it would continuously regenerate electromagnetic field; but this field would continuously fall behind the proton and be lost as radiation, until the proton slowed down to the speed of light, whereupon the electromagnetic field could keep up with it. This is what happens when a medium, such as water, a crystal, or the air, modifies the speed that light travels; charged particles which travel faster radiate "Cherenkov" light.
Ann Nelson and I wrote a paper showing that the same phenomenon occurs when the speed of gravity is slower than the maximum speed of particles. In this case, a high energy particle outruns its own gravitational field. This leads to gravitational radiation, which reduces the particle's energy. The slower the gravitational propagation speed, the faster a high energy particle would radiate away its energy. Since cosmic rays of stunningly high energy arrive at the Earth from astronomical distances (you can find a review article about this here), we can place severe limits on this possibility. Since this paper, Cliff Burgess, Jim Cline, Elise Filotas, Joaquim Matias, and I have written a paper, considering the possibility of quantum interactions conveying a speed difference between gravity and other interactions into a speed difference between different observable particles, and used this to place additional, but model dependent, limits on the propagation speed of gravity.
The very best limits on different propagation speeds for different particle types should arise from the highest energy cosmic rays. The fact that they arrive at the Earth tells us that they do not efficiently radiate away their energy into any other kind of particle. Since particle physics contains a large number of particles, this allows a large number of constraints.
The complication is that the highest energy particles present in the cosmic rays are hadrons, and hadrons are complicated composite objects, made (primarily) out of quarks and gluons held together by the strong force. The exact division between gluons and the different sorts of quarks also depends on the energy scale at which a hadron is analyzed, and it is important to take into account the small admixture in a hadron of particles other than quarks and gluons. For more details, see here. The upshot is that it is possible to set a number of very tight constraints on Lorentz violation, involving most of the particle types of the Standard Model.