  In this paper we prove that the set of $S$-integral points of the smooth
cubic surfaces in $\mathbb{A}^3$ over a number field $k$ is not thin, for
suitable $k$ and $S$. As a corollary, we obtain results on the complement in
$\mathbb{P}^2$ of a smooth cubic curve, improving on Beukers' proof that the
$S$-integral points are Zariski dense, for suitable $S$ and $k$. With our
method we reprove Zariski density, but our result is more powerful since it is
a stronger form of Zariski density. We moreover prove that the rational integer
points on the Fermat cubic surface $x^3+y^3+z^3=1$ form a non-thin set and we
link our methods to previous results of Lehmer, Miller-Woollett and Mordell.
