Take a set $K\subset\mathbb{R}^2$ that is homeomorphic to the Cantor set and has positive $2$-dimensional Lebesgue measure. Take a homeomorphism of the ternary Canor set onto $K$ and extend it as a piecewise $C^\infty$ curves in the complement of the ternary Cantor set. If you make it carefully, you will obtain an injective curve whose image has positive 2-dimensional Lebesgue measure and which is $C^\infty$ smooth in the complement of the set of measure zero - in the complement of the ternary Cantor set.
In higher dimensions $n\geq 2$, it is possible to construct an embedding $f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that it is $C^\infty$ smooth outside a compact set $K\subset\mathbb{S}^n$ of Hausdorff dimension zero and so that $f(\mathbb{S}^n)$ has positive $(n+1)$-dimensional Lebesgue measure.