  We show that Nikodym sets and local smoothing estimates for linear wave
equations form a dichotomy: If Nikodym sets for a family of curves exist, then
the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any
$p<\infty$; if Nikodym sets do not exist, then local smoothing estimates hold,
and the related maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some
$p<\infty$. Whenever the maximal operator is bounded on $L^p(\mathbb{R}^2)$ for
some $p<\infty$, we also determine the sharp exponent for $L^p(\mathbb{R}^2)$
bounds.
