  We apply upper and lower compensated convex transforms, which are `tight'
one-sided approximations of a given function, to the extraction of fine
geometric singularities from semiconvex/semiconcave functions and DC-functions
in $\mathbb{R}^n$ (difference of convex functions). Well-known examples of
(locally) semiconcave functions include the Euclidean distance and squared
distance functions. For a locally semiconvex function $f$ with general modulus,
we show that `locally' a point is a singular (non-differentiable) point if and
only if it is a scale $1$-valley point, and if $x$ is a singular point, then
locally the limit of the scaled valley transform exists at every point $x$ and
$ \lim_{\lambda\to \infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the
radius of the minimal bounding sphere of the (Fr\'echet) subdifferential
$\partial_- f(x)$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus
the limit function $\mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda
V_\lambda (f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of the
singular set for a locally semiconvex function $f$, and also provides an
asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda
\to \infty$. For a locally semiconvex function $f$ with linear modulus we show
that the limit of the gradient of the upper compensated convex transform
$\lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre
of the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function
$f=g-h$, the scale $1$-edge transform satisfies
$\liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq
(r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the
minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_-
h$ of the convex functions $g$ and $h$ at $x$ respectively.
