  We study the approximative trace for individual elements in the Sobolev space
$W^{1,p}(\Omega)$ for $1\le p\le\infty$. This notion of a trace was introduced
for $p=2$ in [AtE11] in the setting of general open sets
$\Omega\subset\mathbb{R}^d$. The approximative trace exhibits a curious
nonuniqueness phenomenon. We provide a detailed analysis of this phenomenon
based on methods of geometric measure theory and are able to give very weak
geometric conditions that are sufficient for the uniqueness of the
approximative trace. In particular, we prove that the approximative trace is
unique on open sets with continuous boundary and on arbitrary connected domains
in $\mathbb{R}^2$. Furthermore, we provide an example which shows that the
uniqueness of the approximative trace depends on $p$. These results answer
several open questions.
