It is known that every $G_\delta$ subset $E$ of the plane containing a dense set of lines, even if it has measure zero,
has the property that every real-valued Lipschitz function on
$\mathbb{R}^2$ has a point of differentiability in $E$ . Here
we show that the set of points of differentiability of
Lipschitz functions inside such sets may be surprisingly tiny:
we construct a $G_\delta$ set $E\subset\mathbb{R}^2$ containing a dense set of lines for which there is a pair of
real-valued Lipschitz functions on $\mathbb{R}^2$ having no
common point of differentiability in $E$ , and there is a
real-valued Lipschitz function on $\mathbb{R}^2$ whose set of
points of differentiability in $E$ is uniformly purely unrectifiable.