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| −2α dθ 2 , where x ∈ R, θ ∈ T and the parameter α ∈ R. For α ≤ −1 this metric describes cone-like manifolds (for α = −1 it is a flat cone). For α = 0 it is a cylinder. For α ≥ 1 it is a Grushin-like metric. We show that the Laplace-Beltrami operator ∆ is essentially self-adjoint if and only if α / ∈ (−3, 1). In this case the only self-adjoint extension is the Friedrichs extension ∆ F , that does not allow communication through the singular set {x = 0} both for the heat and for a quantum particle. For α ∈ (−3, −1] we show that for the Schrödinger equation only the average on θ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ∆ F ) cannot. For α ∈ (−1, 1) we prove that there exists a canonical self-adjoint extension ∆ 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 ∆ F and ∆ B , proving that ∆ F is stochastically complete at the singularity if and only if α ≤ −1, while ∆ B is always stochastically complete at the singularity.