I know that, given an open set $\Omega\subseteq\mathbb{R}^n$, $C_c^{\infty}(\Omega)$ (smooth functions with compact support) is dense in $L^p(\Omega)$, $1\leq p<\infty$.
Let $M$ be a smooth $n-1$ regular surface in $\mathbb{R}^n$, and let $d\sigma$ be the surface measure. Is it true that $C_c^{\infty}(\mathbb{R}^n)$ is dense in $L^p(M,d\sigma)$, $1\leq p<\infty$? That is, if $\int_M |f|^p\,d\sigma<\infty$, can we find $\{f_m\}\subseteq C_c^{\infty}(\mathbb{R}^n)$ such that $\lim_m \int_M|f-f_m|^p\,d\sigma=0$?
If not, which spaces would be dense in $L^p(M,d\sigma)$?
Let us show that $C_c ^\infty (M)$ is dense in $L^p (M)$ ($1 \le p < \infty$), in two steps.
First, $C_c ^\infty (M)$ is dense in $C_0 (M)$ (the space of functions that vanish at infinity) in the topology of compact convergence, by one of the many variations of the Stone-Weierstrass theorem. Since $C_c(M) \subseteq C_0 (M)$, it follows that $C_c ^\infty (M)$ is dense in $C_c (M)$ too.
Next, it is a known result that $C_c (M)$ is dense in $L^p (M)$ (this is true at least for $\sigma$-finite spaces, not only for smooth manifolds).
Combining the two facts you get that $C_c ^\infty (M)$ is dense in $L^p (M)$.