I will mention two diffenent applications:
Uniqueness of the closest point.
Let $K\subset\mathbb{R}^n$ be a compact set. For $x\in \mathbb{R}^n\setminus K$ let $$ D_x=\{y\in K:\, d(x,y)=\operatorname{dist}(x,K)\}. $$ If the set $D_x$ consists of one point, then it means there there is a unique point in $K$ that is closes to $x$. Unfortunately there might be points where the closest point is not unique i.e. when $D_x$ contains more than one point. For example if $K$ is a sphere centered at $x$, then $D_x=K$. However, the set of non-uniqueness points is small:
Theorem. The set of points $x\in \mathbb{R}^n\setminus K$ such that the closest point in $K$ to $x$ is not unique (i.e. $D_x$ has more than one point) has measure zero.
Proof. Indeed, the distance function is Lipschitz and hence differentiable almost everywhere (by Rademacher). However, if the distance is differentiable at $x$, then $D_x$ consists of one point. For a proof of this fact see https://mathoverflow.net/a/299066/121665. $\Box$
Non-embedding of the Heisenberg group.
The Heisenberg group $\mathbb{H}^n$ does not admit a bi-Lipschitz embedding not any Euclidean space. That was observed by Pansu and Semmes and it follows from a version of the Rademacher functions on the Heisenberg group proved by Pansu, see [1] page 397. See also Non embedding of the Heisenberg group
[1] S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights. Rev. Mat. Iberoamericana 12 (1996), 337-410.