  In this paper, we show that for any integer $k \in \mathbb{N}$ there exists a
Sobolev sheaf (in the sense of Lebeau) on any definable site of $\mathbb{R}^2$
that agrees with Sobolev spaces on cuspidal domains. We also provide a complete
computation of the cohomology of these sheaves using the notion of 'Good
direction' introduced by Valette. This paper serves as an introduction to a
more general project on the sheafification of Sobolev spaces in higher
dimensions.
