We study the evolution of the heat and of a free quantum particle (described by the Schrödinger 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 Schrödinger 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.