Scattering, interpreted broadly is just "probe with wave or particle" and "measure response" and thus I claim that it is the principle behind most if not all experimental techniques in physics.
Most of the scattering techniques I give below are "just" applications of Rayleigh scattering, but I hope you can see that the simplicity of Rayleigh scattering is precisely what makes them useful experimental tools!
X-ray and neutron scattering experiments are used to find the structure of complicated materials, for instance. The keyword here is "<PERSON>'s law" in the case of crystal lattices.
A popular technique to probe dynamics is "Dynamic light scattering". The principle here is that scattering can detect fluctuations due to Brownian motion in complex fluids, and this information is encoded in the autocorrelation of the measured intensity.
With scattering experiments, you might think you're screwed if your probe scatters multiple times before it comes back to your detector (because then how will you be able to interpret the data). However, Diffusing wave spectroscopy is a very clever technique that applies in the presence of multiple scattering events.
The field of acoustics is all about scattering of sound waves. In materials there's a lot of interest in studying phonons - these are also probed via scattering of various types.
In the generalized interpretation of scattering, conductivity / resistivity measurements in hard condensed matter also become scattering experiments. 795
There you probe with a electromagnetic wave by causing the voltage in a wire to your sample to oscillate, and you measure the response in the form of electron current. Examples of recently popular materials studied this way: high-temperature superconductors, integer and fractional quantum Hall devices, graphene, topological insulators, etc.
And obviously, from this point of view, particle colliders and detectors do a kind of scattering experiment as well.
The list goes on and on. Many, if not most things studied in physics are waves of some sort, and thus scattering processes are very relevant to experiments. One prominent non-example is most of astrophysics, where we have no chance of actually probing the stars ourselves and can only observe what they do on their own. You might argue that astrophysics is therefore just "observational" and not "experimental". 70