How can they prove the superposition of particle states prior to measurement? If every time a particle's spin or momentum is measured, it gives a discrete answer (collapse of possibility states), how can they ever prove that prior to measurement it was in fact in a super-position of states? Is this solely a logical extrapolation from the wave-like interference patterns seen in the slit experiment? 
Clearly I don't understand something fundamental here. 
 A: There is an approach called weak measurement that can be used to probe the properties of a superposition without destroying it.
There is a reasonable detailed article on it on Wikipedia, or a more accessible article on the Nature web site.
A: One can prepare a lot of the same quantum states and do the same measurement to them, then he will get a series of results. From the this results one can know the possibilities of getting different results, then one can know what the initial quantum state is like. But sometime it cannot be told that whether the initial state is pure or mixed.
A: 
How can they prove the superposition of particle states prior to measurement

Physics theories are not subject to proofs, they are subject to validation of falsification.
One need not prove that the mathematical function describing a measurement for a point (x,y,z,t) describes all of phase space because such is  the construction of the mathematics . This construct predicts a measurement and the hypothesis that the function describes all of phase space is validated because there has been no measurement to falsify it. It is a matter of "trusting" on the truth value of mathematics. The mathematical construct "superposition of states" fits the observable data in innumerable experiments in the microcosm of quantum mechanical solutions.
There is no problem of fitting a parabola to a balistic track, and from measuring  its velocity  direction and mass extrapolate to its origin  in (x,y,z,t). Gravitational laws have been validated innumerable times. Similarly, quantum mechanical description of data have been validated innumerable times. The concepts are more complicated, but the trust in mathematics the same. 
A: You don't prove that a specific qubit is in superposition, you prove that a specific method of preparing qubits creates qubits in superposition and then infer.
For example, suppose you have qubit-preparing method X. You run it many, many times and produce a huge number of qubits. You measure those qubits in all different directions, checking what the measurement biases are. You confirm those biases are consistent over time and lots of other variables, and find that they match the biases predicted by the state $\frac{1}{\sqrt 2}|0\rangle + \frac{1}{\sqrt 2}|1\rangle$.
You find lots of other processes that all fit into the mold of quantum-mechanics. You gain confidence. You start thinking in terms of the properties of each qubit, instead of the properties of the processes that produced the qubit.
Eventually, after much checking of notes, you declare that method X reliably produces qubits in the state $\frac{1}{\sqrt 2}|0\rangle + \frac{1}{\sqrt 2}|1\rangle$. There is no other simple explanation known for all the tests you've done, and many theorems that show whole classes of simpler explanations wouldn't work.
A: You question involves a misrepresentation of how science works. Science doesn't prove anything. Rather, scientists make progress by doing the following, as explained by Karl Popper (see "Realism and the Aim of Science", Chapter I). (If they don't do what I'm describing they don't make progress.) (1) They propose ideas to try to solve problems with current theories. (2) They then look for problems with their proposals, such as failing to solve the problem they were created to solve or clashing with a new experimental result. (3) Once they have a theory that solves the original problem and doesn't have any outstanding problems they start to look for problems with the new idea.
So then the question that should be asked is, if there is a superposition of states before measurement how could that idea be tested? What testable statement would have to be true if there is a superposition that would not be true if there was no superposition? There are many experiments that do such tests. If you do a single particle interference experiment in an interferometer and put a phase shifter in one arm of the interferometer that tests whether there was a superposition.
