Since non-abelian anyons have become quite fashionable from the point of view of theory. I would like to know, whether there has actually been experimental confirmation of such objects. If you could cite original literature, that would be great!
As far as I know we do not yet have definitive verification of non-abelian statistics which would indicate the existence of non-abelian Anyons. The latest results I know about are the ones mentioned by akhmeteli and An, et. al. "Braiding of Abeliana and Non-Abelian Anyons in the Fractional Quantum Hall Effect." Arxiv:1112.3400. However, I do not believe either are accepted as definitive proof.
You are correct in that there are other candidate systems proposed, such as $p+ip$ superfluids, superconductors, and other systems. Some good references and also a review seem to be given in these review articles. As I understand it, the big push to observe evidence of Majorana fermions. That we do have some pretty good evidence for. Pairs of Majorana fermions are supposed to realize non-abelian statistics, but this has not yet been implemented and observed (to my knowledge).
Try http://arxiv.org/abs/1301.2639 (Phys. Rev. Lett. 111, 186401 (2013)): "Magnetic field-tuned Aharonov-Bohm oscillations and evidence for non-Abelian anyons at v=5/2" , although I am not sure this is a definite evidence.
"We show that the resistance of the v=5/2 quantum Hall state, confined to an interferometer, oscillates with magnetic field consistent with an Ising-type non-Abelian state."