In this paper we give the first examples of positive closed currents in
$\mathbb{C}^2$ with continuous potentials, vanishing self-intersection, and
which are not laminar. More precisely, they are supported on sets "without
analytic structure". The result is mostly interesting when the potential has
regularity close to $C^2$, because laminarity is expected to hold in that case.
We actually construct examples which are $C^{1,\alpha}$ for all $\alpha<1$.