  In this note, we showcase some recent results concerning the stickiness
properties of nonlocal minimal graphs in the plane. To start with, the nonlocal
minimal graphs in the plane enjoy an enhanced boundary regularity, since
boundary continuity with respect to the external datum is sufficient to ensure
differentiability across the boundary of the domain.
  As a matter of fact, the H\"older exponent of the derivative is in this
situation sufficiently high to provide the validity of the Euler-Lagrange
equation at boundary points as well.
  From this, using a sliding method, one also deduces that the stickiness
phenomenon is generic for nonlocal minimal graphs in the plane, since an
arbitrarily small perturbation of continuous nonlocal minimal graphs can
produce boundary discontinuities (making the continuous case somehow
"exceptional" in this framework).
