  In this paper we study Hardy-Sobolev inequalities on hypersurfaces of
$\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having
universal constants independent of the hypersurface. We first consider the
celebrated Sobolev inequality of Michael-Simon and Allard, in our codimension
one framework. Using their ideas, but simplifying their presentations, we give
a quick and easy-to-read proof of the inequality. Next, we establish two new
Hardy inequalities on hypersurfaces. One of them originates from an application
to the regularity theory of stable solutions to semilinear elliptic equations.
The other one, which we prove by exploiting a "ground state" substitution,
improves the Hardy inequality of Carron. With this same method, we also obtain
an improved Hardy or Hardy-Poincar\'e inequality.
