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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 $f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ that it is $C^\infty$ smooth outside a compact set $C\subset\mathbb{S}^n$ of Hausdorff dimension zero and such that $f(\mathbb{S}^n)$ has positive $(n+1)$-dimensional Lebesgue measure.

In fact for any set $K\subset\mathbb{R}^{n+1}$ that is homeomorphic to the ternary Cantor set, one can construct $f$ as above so that $f(C)=K$. Since $K$ can have positive measure, the above result follows. That construction also includes various generalizations of the horned sphere when taking wild Cantor sets as $K$.

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 $f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ that it is $C^\infty$ smooth outside a compact set $C\subset\mathbb{S}^n$ of Hausdorff dimension zero and such that $f(\mathbb{S}^n)$ has positive $(n+1)$-dimensional Lebesgue measure.

In fact for any set $K\subset\mathbb{R}^{n+1}$ that is homeomorphic to the ternary Cantor set, one can construct $f$ as above so that $f(C)=K$. Since $K$ can have positive measure, the above result follows. That construction also includes various generalizations of the horned sphere when taking wild Cantor sets as $K$.

A construction of such an embedding with additional properties can be found in:

P. Hajłasz, X. Zhou, Sobolev homeomorphism on a sphere containing an arbitrary Cantor set in the image. Geom. Dedicata 184 (2016), 159-173. arXiv

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.

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 $f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ that it is $C^\infty$ smooth outside a compact set $C\subset\mathbb{S}^n$ of Hausdorff dimension zero and such that $f(\mathbb{S}^n)$ has positive $(n+1)$-dimensional Lebesgue measure.

In fact for any set $K\subset\mathbb{R}^{n+1}$ that is homeomorphic to the ternary Cantor set, one can construct $f$ as above so that $f(C)=K$. Since $K$ can have positive measure, the above result follows. That construction also includes various generalizations of the horned sphere when taking wild Cantor sets as $K$.

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.

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.