For spin measurements the original experiment was the Stern-Gerlach experiment in which you will see that a prior unpolarized beam will split up in two (Spin up and down) orientations.
For helicity, a very ingenious and fascinating experiment is the famours Goldhaber experiment that uses a very peculiar set of elements and also the Mößbauer effect to measure the helicity of neutrinos. The helicity is the projection of the Spin onto the momentum direction, and thus if you measure spin and momentum, you can compute helicity.
Link to original paper here: http://www.bnl.gov/nh50/
People often confuse helicity and chirality they are only the same in the case of massless particles. This is also the reason why we know that (at least interacting) neutrinos are always left-handed (the famous $$SU(2)_L$$ ), at least if we assume that neutrinos are massless (which is almost the case, but not strictly).
In contrast to helicity for massive particles (where you can always boost into a frame where you change the sign of the momentum direction and thus change helicity), chirality is a Lorentz-invariant property of the particle.