  We prove H\"older regularity of any continuous solution $u$ to a $1$-D scalar
balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the
flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For
example, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost every
point $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to
0$. Due to Lipschitz regularity along characteristics, this implies that at
almost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) =
o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results to
provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions
in the first Heisenberg group.
