First of all, a majorana fermion does not "possess non-abelian statistics", it is predicted to obey non-abelian braiding statistics which may serve as a building block for fault-tolerant quantum computation.
There have been studies that indicate the presence of a Majorana end states for example in semiconductor quantum wires coupled to a conventional supercondcutor, inside vortex cores of superconducting compounds or at the end of 1D Fe chains on Pb. Evidence for such states may be a narrow zero-bias peak in the differential conductance of an STM investigation while excluding alternative explanations such as Shiba states, the Kondo effect or disorder effects.
In order to examine its use in quantum computation however, you would have to realize a quantum computer. And although there a theoretical predictions, nobody seems to have more than a vage idea exactly how to do this.
Experimental evidence has started to emerge only in the last 4-5 years, so there is still a lot of research to do. There are several groups are currently working on the subject, but it my best guress would be that a breakthrough is still not in sight..