  We study the evolution of the heat and of a free quantum particle (described
by the Schr\"odinger equation) on two-dimensional manifolds endowed with the
degenerate Riemannian metric $ds^2=dx^2+|x|^{-2\alpha}d\theta^2$, where $x\in
\mathbb R$, $\theta\in\mathbb T$ and the parameter $\alpha\in\mathbb R$. For
$\alpha\le-1$ this metric describes cone-like manifolds (for $\alpha=-1$ it is
a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha\ge 1$ it is a
Grushin-like metric. We show that the Laplace-Beltrami operator $\Delta$ is
essentially self-adjoint if and only if $\alpha\notin(-3,1)$. In this case the
only self-adjoint extension is the Friedrichs extension $\Delta_F$, that does
not allow communication through the singular set $\{x=0\}$ both for the heat
and for a quantum particle. For $\alpha\in(-3,-1]$ we show that for the
Schr\"odinger equation only the average on $\theta$ of the wave function can
cross the singular set, while the solutions of the only Markovian extension of
the heat equation (which indeed is $\Delta_F$) cannot. For $\alpha\in(-1,1)$ we
prove that there exists a canonical self-adjoint extension $\Delta_B$, called
bridging extension, which is Markovian and allows the complete communication
through the singularity (both of the heat and of a quantum particle). Also, we
study the stochastic completeness (i.e., conservation of the $L^1$ norm for the
heat equation) of the Markovian extensions $\Delta_F$ and $\Delta_B$, proving
that $\Delta_F$ is stochastically complete at the singularity if and only if
$\alpha\le -1$, while $\Delta_B$ is always stochastically complete at the
singularity.
