**I will mention five different applications:**


----------


 - **Characterization of almost everywhere differentiability.**

----------

The following result is a consequence of the Rademacher theorem:

> **Theorem (Stepanov).** A function $f:\Omega\to\mathbb{R}$ defined on an open set $\Omega\subset\mathbb{R}^n$ is fifferentiable almost
> everywhere if and only if $$ \lim_{y\to
 x}\frac{|f(y)-f(x)|}{|y-x|}<\infty $$ for almost all $x\in\Omega$.

A beautiful proof of this classical result is given in [4].

----------


 - **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. $\Box$

For a related post see: https://mathoverflow.net/q/342308/121665.


----------

 - **Non-embedding of the Heisenberg group.**


----------


> **Theorem (Semmes).** The Heisenberg group $\mathbb{H}^n$ does not admit a bi-Lipschitz embedding not any Euclidean space.

That was observed by Semmes and it follows from a version of the Rademacher functions on the Heisenberg group proved by Pansu, see [1] page 397. 
See also https://mathoverflow.net/q/297806/121665 


----------

 - **Kirchheim-Rademacher theorem.**

----------


Suppose that $f:\Omega\to\mathbb{R}^m$ (not necessarily Lipschitz) is differentiable at $x\in\Omega$. Then
$$
\left|\frac{|f(y)-f(x)|-|Df(x)(y-x)|}{|y-x|}\right| \leq
\frac{|f(y)-f(x)-Df(x)(y-x)|}{|y-x|} \stackrel{y\to x}{\longrightarrow} 0.
$$
Observe that $\Vert z\Vert_x:=|Df(x)z|$ is a 
seminorm (that mens $\Vert z_1+z_2\Vert_x\leq\Vert z_1\Vert_x+\Vert z_2\Vert_x$,
$\Vert tz\Vert_x=|t|\Vert z\Vert_x$, but $\Vert\cdot\Vert_x$ may vanish on a subspace of $\mathbb{R}^n$).

> **Theorem (Kirchheim).** If $f:\mathbb{R}^n\supset\Omega\to X$ is a Lipschitz continuous mapping into any metric space $(X,d)$, then for
 almost all $x\in\Omega$, there is a seminorm $\Vert\cdot\Vert_x$ in
 $\mathbb{R}^n$ such that $$ \frac{d(f(y),f(x))-\Vert y-x\Vert_x}{|y-x|}
 \to 0 \quad \mbox{as $y\to x$.} $$

The seminorm $\Vert\cdot\Vert_x$ is called the *metric derivative* of $f$.

This result was proved in [2]. For another proof see [1].


----------

 - **Length preserving mappings**

----------

(See also the answer of Anton Petrunin).
In  [5], Theorem 2.4.11, Gromov proved that any Riemannian manifold of dimension $n$ admits a mapping into 
$\mathbb{R}^n$ that preserves lengths of curves. 
Such a mapping is necessarily Lipschitz and hence differentiable almost everywhere (by Rademacher). One can prove that the Jacobian of such a mapping is different than zero almost everywhere, 
and hence there is no curve-length preserving mapping into $\mathbb{R}^m$ for $m < n$. 


----------

References
----------




**[1]  L. Ambrosio, B. Kirchheim,** Bernd Rectifiable sets in metric and Banach spaces. *Math. Ann.* 318 (2000), 527–555.

**[2] B. Kirchheim,** 
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. *Proc. Amer. Math. Soc.* 121 (1994), 113-123. 

**[3] S. Semmes,** On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights. *Rev. Mat. Iberoamericana* 12 (1996), 337-410. 

**[4]  J. Maly,** A simple proof of the Stepanov theorem on differentiability almost everywhere. *Exposition. Math.* 17 (1999), no. 1, 59–61.

**[5] M. Gromov,**
*Partial differential relations.*
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 
[Results in Mathematics and Related Areas (3)], 9. 
Springer-Verlag, Berlin, 1986.