Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is $n$-rectifiable) but for which there is not locally bi-Lipschitz equivalent to a subset of $\mathbb{R}^n$? The example should be topologically simple, and as "sparse" as possible.

Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is $n$-rectifiable) but which is not locally bi-Lipschitz equivalent to a subset of $\mathbb{R}^n$? The example should be topologically simple, and as "sparse" as possible.

Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is $n$-rectifiable) but for which there is no bi-Lipschitz map onto a subset of $\mathbb{R}^n$?

Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is $n$-rectifiable) but for which there is not locally bi-Lipschitz equivalent to a subset of $\mathbb{R}^n$? The example should be topologically simple, and as "sparse" as possible.