The following application may not be too far outside the realm of usual applications, as described in the OP's original question statement, but I will let others decide for themselves.

Rademacher's theorem is used to prove a maximum/minimum principle for 2D drift-diffusion equations $$\partial_t\theta + u\cdot\nabla\theta + \kappa(-\Delta)^{\gamma/2}\theta, \tag{1}$$ where $\theta:\mathbb{R}^2\rightarrow\mathbb{R}$ is a scalar-valued function, $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a divergence-free vector field possibly depending on $\theta$, $\kappa\geq 0$, and $(-\Delta)^{\gamma/2}$ is the fractional Laplacian of order $\gamma$. In the case where the velocity field $u$ is recovered from $\theta$ through the Biot-Savart law $$u = (-R_2\theta,R_1\theta),$$ where $R_1,R_2$ are the usual 2D Riesz transforms, we recover the well-known dissipative Surface Quasi-Geostrophic (SQG) equation.

When $\kappa=0$, it is easy to see from the existence of a flow map that the minimum and maximum of a smooth solution $\theta$ are conserved. For $\kappa>0$, this is no longer necessarily the case; however, the minimum is nondecreasing and the maximum is nonincreasing. To see this, one argue as follows, as for instance shown in this paper by the Cordobas. Denote the supremum $\theta$ at time $t$ respectively by $$M(t) := \sup_{x\in\mathbb{R}^2} \theta(t,x).$$ If $\theta(0)\in L^\infty$, then since $\|\theta(t)\|_{L^p} \leq \|\theta(0)\|_{L^p}$, for any $1\leq p\leq \infty$, (see the aforementioned paper), it follows that $M(t)$ is finite. Moreover, if $\theta(t)\in \hat{L}^1$ for each $t$, then it follows from the Riemann-Lebesgue lemma that $\theta(t)\in C_0$. Hence, for each $t$, there exists a point $x(t)\in\mathbb{R}^2$ such that $$M(t) = \theta(t,x(t)).$$ It's not hard to show that $M(t)$ is a Lipschitz continuous function. Hence by Rademacher's theorem, $M(t)$ is differentiable for almost every $t$. At such a point $t_0$ of differentiability, we can find by a compactness argument a sequence of positive numbers $h_j\rightarrow 0$ such that $x(t_0+h_j)\rightarrow x_0\in\mathbb{R}^2$ as $j\rightarrow\infty$. Since $M$ is continuous, we then have $M(t_0) = \theta(t_0,x_0)$. Now \begin{align} \frac{M(t_0+h_j)-M(t_0)}{h_j} &= \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &=\frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j} + \frac{\theta(t_0,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &\leq \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j}, \end{align} since the second term in the penultimate line is nonpositive. Letting $j\rightarrow\infty$ and using that $t_0$ is a point of differentiability by assumption, we obtain that $$M'(t_0) \leq \partial_t\theta(t_0,x_0) = -u(t_0,x_0) \cdot \nabla\theta(t_0,x_0) - \kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0).$$ The first term on the RHS of the preceding equality is nonpositive since $\theta(t_0)$ is maximal at $x_0$. The first term on the RHS is also nonpositive for the same reason since $$(-\Delta)^{\gamma/2}\theta(t_0,x_0) = C_\gamma PV\int_{\mathbb{R}^2}\frac{\theta(t_0,x_0)-\theta(t_0,y)}{|x-y|^{2+\gamma}}dy.$$ Hence, $M'(t_0)\leq 0$. Note that by considering a sequence $h_j$ of negative numbers, one can show that $M'(t_0) = -\kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0)$. That $m$ is nondecreasing follows by replacing $\theta$ by $-\theta$ in the preceding argument.

The following application may not be too far outside the realm of usual applications, as described in the OP's original question statement, but I will let others decide for themselves.

Rademacher's theorem is used to prove a maximum/minimum principle for 2D drift-diffusion equations $$\partial_t\theta + u\cdot\nabla\theta + \kappa(-\Delta)^{\gamma/2}\theta, \tag{1}$$ where $\theta:\mathbb{R}^2\rightarrow\mathbb{R}$ is a scalar-valued function, $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a divergence-free vector field possibly depending on $\theta$, $\kappa\geq 0$, and $(-\Delta)^{\gamma/2}$ is the fractional Laplacian of order $\gamma$. In the case where the velocity field $u$ is recovered from $\theta$ through the Biot-Savart law $$u = (-R_2\theta,R_1\theta),$$ where $R_1,R_2$ are the usual 2D Riesz transforms, we recover the well-known dissipative Surface Quasi-Geostrophic (SQG) equation.

When $\kappa=0$, it is easy to see from the existence of a flow map that the minimum and maximum of a smooth solution $\theta$ are conserved. For $\kappa>0$, this is no longer necessarily the case; however, the minimum is nondecreasing and the maximum is nonincreasing. To see this, one argue as follows, as for instance shown in this paper by the Cordobas. Denote the supremum $\theta$ at time $t$ respectively by $$M(t) := \sup_{x\in\mathbb{R}^2} \theta(t,x).$$ If $\theta(0)\in L^\infty$, then since $\|\theta(t)\|_{L^p} \leq \|\theta(0)\|_{L^p}$, for any $1\leq p\leq \infty$, (see the aforementioned paper), it follows that $M(t)$ is finite. Moreover, if $\theta(t)\in \hat{L}^1$ for each $t$, then it follows from the Riemann-Lebesgue lemma that $\theta(t)\in C_0$. Hence, for each $t$, there exists a point $x(t)\in\mathbb{R}^2$ such that $$M(t) = \theta(t,x(t)).$$ It's not hard to show that $M(t)$ is a Lipschitz continuous function. Hence by Rademacher's theorem, $M(t)$ is differentiable for almost every $t$. At such a point $t_0$ of differentiability, we can find by a compactness argument a sequence of positive numbers $h_j\rightarrow 0$ such that $x(t_0+h_j)\rightarrow x_0\in\mathbb{R}^2$ as $j\rightarrow\infty$. Since $M$ is continuous, we then have $M(t_0) = \theta(t_0,x_0)$. Now \begin{equation*} \begin{split} &\frac{M(t_0+h_j)-M(t_0)}{h_j} = \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &=\frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j} + \frac{\theta(t_0,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &\leq \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j}, \end{split} \end{equation*} since the second term in the penultimate line is nonpositive. Letting $j\rightarrow\infty$ and using that $t_0$ is a point of differentiability by assumption, we obtain that $$M'(t_0) \leq \partial_t\theta(t_0,x_0) = -u(t_0,x_0) \cdot \nabla\theta(t_0,x_0) - \kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0).$$ The first term on the RHS of the preceding equality is nonpositive since $\theta(t_0)$ is maximal at $x_0$. The first term on the RHS is also nonpositive for the same reason since $$(-\Delta)^{\gamma/2}\theta(t_0,x_0) = C_\gamma PV\int_{\mathbb{R}^2}\frac{\theta(t_0,x_0)-\theta(t_0,y)}{|x-y|^{2+\gamma}}dy.$$ Hence, $M'(t_0)\leq 0$. Note that by considering a sequence $h_j$ of negative numbers, one can show that $M'(t_0) = -\kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0)$. That $m$ is nondecreasing follows by replacing $\theta$ by $-\theta$ in the preceding argument.

The following application may not be too far outside the realm of usual applications, as described in the OP's original question statement, but I will let others decide for themselves.

Rademacher's theorem is used to prove a maximum/minimum principle for 2D dissipation drift-diffusion equations $$\partial_t\theta + u\cdot\nabla\theta + \kappa(-\Delta)^{\gamma/2}\theta, \tag{1}$$ where $\theta:\mathbb{R}^2\rightarrow\mathbb{R}$ is a scalar-valued function, $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a divergence-free vector field possibly depending on $\theta$, $\kappa\geq 0$, and $(-\Delta)^{\gamma/2}$ is the fractional Laplacian of order $\gamma$. In the case where the velocity field $u$ is recovered from $\theta$ through the Biot-Savart law $$u = (-R_2\theta,R_1\theta),$$ where $R_1,R_2$ are the usual 2D Riesz transforms, we recover the well-known dissipative Surface Quasi-Geostrophic (SQG) equation.

When $\kappa=0$, it is easy to see from the existence of a flow map that the minimum and maximum of a smooth solution $\theta$ are conserved. For $\kappa>0$, this is no longer necessarily the case; however, the minimum is nondecreasing and the maximum is nonincreasing. To see this, one argue as follows, as for instance shown in this paper by the Cordobas. Denote the supremum $\theta$ at time $t$ respectively by $$M(t) := \sup_{x\in\mathbb{R}^2} \theta(t,x).$$ If $\theta(0)\in L^\infty$, then since $\|\theta(t)\|_{L^p} \leq \|\theta(0)\|_{L^p}$, for any $1\leq p\leq \infty$, (see the aforementioned paper), it follows that $M(t)$ is finite. Moreover, if $\theta(t)\in \hat{L}^1$ for each $t$, then it follows from the Riemann-Lebesgue lemma that $\theta(t)\in C_0$. Hence, for each $t$, there exists a point $x(t)\in\mathbb{R}^2$ such that $$M(t) = \theta(t,x(t)).$$ It's not hard to show that $M(t)$ is a Lipschitz continuous function. Hence by Rademacher's theorem, $M(t)$ is differentiable for almost every $t$. At such a point $t_0$ of differentiability, we can find by a compactness argument a sequence of positive numbers $h_j\rightarrow 0$ such that $x(t_0+h_j)\rightarrow x_0\in\mathbb{R}^2$ as $j\rightarrow\infty$. Since $M$ is continuous, we then have $M(t_0) = \theta(t_0,x_0)$. Now \begin{align} \frac{M(t_0+h_j)-M(t_0)}{h_j} &= \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &=\frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j} + \frac{\theta(t_0,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &\leq \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j}, \end{align} since the second term in the penultimate line is nonpositive. Letting $j\rightarrow\infty$ and using that $t_0$ is a point of differentiability by assumption, we obtain that $$M'(t_0) \leq \partial_t\theta(t_0,x_0) = -u(t_0,x_0) \cdot \nabla\theta(t_0,x_0) - \kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0).$$ The first term on the RHS of the preceding equality is nonpositive since $\theta(t_0)$ is maximal at $x_0$. The first term on the RHS is also nonpositive for the same reason since $$(-\Delta)^{\gamma/2}\theta(t_0,x_0) = C_\gamma PV\int_{\mathbb{R}^2}\frac{\theta(t_0,x_0)-\theta(t_0,y)}{|x-y|^{2+\gamma}}dy.$$ Hence, $M'(t_0)\leq 0$. Note that by considering a sequence $h_j$ of negative numbers, one can show that $M'(t_0) = -\kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0)$. That $m$ is nondecreasing follows by replacing $\theta$ by $-\theta$ in the preceding argument.

The following application may not be too far outside the realm of usual applications, as described in the OP's original question statement, but I will let others decide for themselves.

Rademacher's theorem is used to prove a maximum/minimum principle for 2D drift-diffusion equations $$\partial_t\theta + u\cdot\nabla\theta + \kappa(-\Delta)^{\gamma/2}\theta, \tag{1}$$ where $\theta:\mathbb{R}^2\rightarrow\mathbb{R}$ is a scalar-valued function, $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a divergence-free vector field possibly depending on $\theta$, $\kappa\geq 0$, and $(-\Delta)^{\gamma/2}$ is the fractional Laplacian of order $\gamma$. In the case where the velocity field $u$ is recovered from $\theta$ through the Biot-Savart law $$u = (-R_2\theta,R_1\theta),$$ where $R_1,R_2$ are the usual 2D Riesz transforms, we recover the well-known dissipative Surface Quasi-Geostrophic (SQG) equation.

When $\kappa=0$, it is easy to see from the existence of a flow map that the minimum and maximum of a smooth solution $\theta$ are conserved. For $\kappa>0$, this is no longer necessarily the case; however, the minimum is nondecreasing and the maximum is nonincreasing. To see this, one argue as follows, as for instance shown in this paper by the Cordobas. Denote the supremum $\theta$ at time $t$ respectively by $$M(t) := \sup_{x\in\mathbb{R}^2} \theta(t,x).$$ If $\theta(0)\in L^\infty$, then since $\|\theta(t)\|_{L^p} \leq \|\theta(0)\|_{L^p}$, for any $1\leq p\leq \infty$, (see the aforementioned paper), it follows that $M(t)$ is finite. Moreover, if $\theta(t)\in \hat{L}^1$ for each $t$, then it follows from the Riemann-Lebesgue lemma that $\theta(t)\in C_0$. Hence, for each $t$, there exists a point $x(t)\in\mathbb{R}^2$ such that $$M(t) = \theta(t,x(t)).$$ It's not hard to show that $M(t)$ is a Lipschitz continuous function. Hence by Rademacher's theorem, $M(t)$ is differentiable for almost every $t$. At such a point $t_0$ of differentiability, we can find by a compactness argument a sequence of positive numbers $h_j\rightarrow 0$ such that $x(t_0+h_j)\rightarrow x_0\in\mathbb{R}^2$ as $j\rightarrow\infty$. Since $M$ is continuous, we then have $M(t_0) = \theta(t_0,x_0)$. Now \begin{align} \frac{M(t_0+h_j)-M(t_0)}{h_j} &= \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &=\frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j} + \frac{\theta(t_0,x(t_0+h_j))-\theta(t_0,x_0)}{h_j} \\ &\leq \frac{\theta(t_0+h_j,x(t_0+h_j))-\theta(t_0,x(t_0+h_j))}{h_j}, \end{align} since the second term in the penultimate line is nonpositive. Letting $j\rightarrow\infty$ and using that $t_0$ is a point of differentiability by assumption, we obtain that $$M'(t_0) \leq \partial_t\theta(t_0,x_0) = -u(t_0,x_0) \cdot \nabla\theta(t_0,x_0) - \kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0).$$ The first term on the RHS of the preceding equality is nonpositive since $\theta(t_0)$ is maximal at $x_0$. The first term on the RHS is also nonpositive for the same reason since $$(-\Delta)^{\gamma/2}\theta(t_0,x_0) = C_\gamma PV\int_{\mathbb{R}^2}\frac{\theta(t_0,x_0)-\theta(t_0,y)}{|x-y|^{2+\gamma}}dy.$$ Hence, $M'(t_0)\leq 0$. Note that by considering a sequence $h_j$ of negative numbers, one can show that $M'(t_0) = -\kappa(-\Delta)^{\gamma/2}\theta(t_0,x_0)$. That $m$ is nondecreasing follows by replacing $\theta$ by $-\theta$ in the preceding argument.