Is it entirely theoretical, or what?
From Wikipedia:
Abelian anyons have been detected and play a major role in the fractional quantum Hall effect.
So, no - it's detected. However, the non-Abelian are just theorized. From Wikipedia:
Non-abelian anyons have not been definitively detected, although this is an active area of research.
I'm unsure of any practical use - but the observation/confirmation could lead to more confidence in the quantum mechanic models and help further the refinement of that field of study.