I had forgotten about this, but a while ago a friend of mine did his whole thesis work on the Rayleigh-Taylor instability in the context of statistical analysis of turbulence. You can find it enclosed here :
The thesis is in French, but the work is extremely thorough concerning the RT instability. You might find the quantitative information (growth rate, turbulent fluctuations, etc.) that you are interested in as well as the problem setup. If you need help with some parts, I can help with the translation.
On another note, if you wish to study the order of convergence of your method (if you wish to study stability, ignore what I am about to say) the best approach to test your convergence is to design test cases using the method of manufactured solution.
Reference book : https://www.amazon.ca/Verification-Validation-Scientific-Computing-Oberkampf/dp/0521113601
Example of a recent article using this (this one is by me, but there are many others) :http://www.sciencedirect.com/science/article/pii/S0045793015000675
Concretly, the method of manufactured solution allows you to create analytical solutions to the incompressible Navier-Stokes equations that are infinitely continuous, that stimulate every term in the Navier-Stokes equation equally and that may even possibly be unsteady. This allows you to fully assess the order of convergence (or the run time, etc.) on fully complex test cases that are representative of the real applications of your CFD code. Furthermore, they can be combined with any type of boundary condition (Dirichlet, Periodic, Neumann, Robin).
That is a good way to benchmark a numerical method and may be even used to evaluate RANS models (with or without wall models). For the latter, see the work of Pr. Pelletier's group.
If this is not what you are looking for, then the Rayleigh-Taylor flow can be good example of an unsteady test case. I will try to look-up a reference.