This is a very intriguing theoretical paper that proposes the detection of anyon using STM (you get free access to the actual paper from the website). The detection involves the measurement of the local density of states (LDOS), and then counting the resonance "rings". This is shown in Fig. 1 and 2 of the paper.
This is quite a fascinating idea, because to get these fractional effects, one has to have a 2D confinement of the charges involved.
Now it becomes a race in seeing who might be able to produce such an experiment to detect these rings. STMs are pretty common, but it is now a matter of having the suitable material to see this.
 Z. Papic et. al. PRX v.8, 011037 (2018).