It's not that simple. See about polar/nonpolar points/sets e.g. in http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Brownian_Motion_and_Harmonic_functions

If I remember correctly, a set is not polar iff it has positive capacity (w.r.t. logarithmic potential in two dimensions, and Newton potential for $d\geq 3$).