Delta function singularity in curvature Are there 3+1D spacetimes that lead to a $\delta$-function in curvature? Are there any examples that one can provide?
 A: A good analysis of this subject is contained in the paper R. Geroch, J. Traschen, Strings and other distributional sources in general relativity.
They conclude that thin shells (delta singularity on a codimension 1 hypersurface) are well-defined but higher codimension surfaces supporting delta singularities in the curvature are not.
I will not be completely rigorous with my use of distributions here, since it takes quite a lot of effort to state how the theory works on manifolds correctly. I will also not be completely general in the formulation in the sense that I will consider timelike hypersurfaces only (the spacelike case is identical with some sign changes here and there), even though with small modifications, the considerations here are also valid for general hypersurfaces with no assumptions on its causal type.
Let $M$ be a smooth four dimensional manifold (four dimensionality is not strictly required though) with a Lorentzian metric tensor $g$ on it and a hypersurface $\Sigma\subseteq M$ such that $g$ is smooth away from $M$ but is merely assumed to be continuous on $\Sigma$. It is further assumed that $\Sigma$ partitions $M$ into two disconnected subdomains $M_+$ and $M_-$ with $\Sigma$ being the common boundary surface separating the two.
Note that the induced metric $h=i^\ast g$ ($i:\Sigma\rightarrow M$ is the inclusion) on $\Sigma$ is not only unambigous under these assumptions, but is smooth.
One may then derive (this is the so-called Israel formalism for thin shells) that the curvature tensor exists as a tensor distribution on $M$ in the following form: $$ \mathrm{Riem}=\mathrm{Riem}_+\Theta_\Sigma+\mathrm{Riem}_-\left(1-\Theta_\Sigma\right)+Q\delta_\Sigma, $$ where 1) $$ \Theta_\Sigma(p)=\left\{\begin{matrix} 1 & p\in M_+ \\ 0 & p\in M_- \end{matrix}\right. $$ is the step function associated with the hypersurface and the Dirac delta $\delta_\Sigma$ associated with the hypersurface is defined by $$\langle \delta_\Sigma,\varphi\rangle=\int_\Sigma\varphi\mu_h $$ for any test function $\varphi$ (smooth with compact support) on $M$ with $\mu_h$ being the induced volume form of the metric $h$ on $\Sigma$,  2) $\mathrm{Riem_\pm}$ are the curvature tensors calculated from the smooth metrics $g_{\pm}$ in $M_\pm$ and 3) $Q$ is a smooth tensor field defined only along $\Sigma$.
Letting $(U,x^\mu),\ \mu=0,1,2,3$ be an "admissible" chart (i.e. one which passes through $\Sigma$ smoothly) on $M$ intersecting $\Sigma$, the coefficients $Q_{\kappa\lambda\mu\nu}$ of the tensor field $Q$ are $$ Q_{\kappa\lambda\mu\nu}=n_{\kappa}\left[K_{\lambda\mu}\right]n_{\nu}-n_{\kappa}\left[K_{\lambda\nu}\right]n_{\mu}+n_{\lambda}\left[K_{\kappa\nu}\right]n_{\mu}-n_{\lambda}\left[K_{\kappa\mu}\right]n_{\nu}, $$ where $$ [K_{\mu\nu}](p)=\lim_{q\in M_+\rightarrow p}K^+_{\mu\nu}(q)-\lim_{q\in M_-\rightarrow p}K^{-}_{\mu\nu}(q) $$ is the difference of the extrinsic curvature tensors of $\Sigma$ as calculated from the two sides.
This formalism gives a large class of metrics whose curvature tensors are singular.

I might update this answer with a concrete example in the future, but the book E. Poisson, A relativist's toolkit contains a number of worked out examples (for example Section 3.9).
The example in Section 3.9 considers a timelike hypersurface $\Sigma$ which is spatially spherical with the "interior" region $M_-$ being Minkowski spacetime and the "exterior" region $M_+$ being Schwarzschild spacetime. A singular dust energy-momentum tensor is placed on $\Sigma$. The resulting spacetime is a valid, albeit weak, solution of the Einstein field equations in which the curvature tensor has a Dirac delta singularity.
A: The simplest example I can think of is to have a "bumpy" metric (also here):
\begin{equation}
g_{\mu \nu} = 
\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & 1 + \theta (x - x_{0}) & 0 & 0 \\
0 & 0 & 1 + \theta (x - x_{0}) & 0 \\
0 & 0 & 0 & 1+ \theta (x - x_{0}) \\
\end{pmatrix}
\end{equation}
where $x_{0}$ the proper distance after which the spatial directions of the metric experience a sudden "jump". Any Christoffel symbol involving the derivative $\partial _{\mu} g_{ii} \, , \, i = 1,2,3$ yields a delta function $\delta (x - x_{0})$ times other stuff depending on the derivative at hand. Since some Christoffel symbols now have a delta function, so will the Riemannn curvature tensor:
\begin{equation}
R^{\mu} {}_{ \rho \nu \sigma} = \nabla _{\nu} \Gamma ^{\mu}{}_{\rho \sigma} - \nabla_{\sigma} \Gamma ^{\mu}{}_{\nu \rho}
\end{equation}
where $\nabla _{\rho}$ the covariant derivative (which itself will include a Christoffel symbol). Subsequently the Ricci tensor:
\begin{equation}
R_{\rho \sigma} = R^{\mu}{}_{\rho \mu \sigma}
\end{equation}
will also have components that contain a delta function.
The scalar curvature $R = g^{\rho \sigma} R_{\rho \sigma}$ will carry those delta functions trivially, since the inverse metric $g^{\rho \sigma}$ will have the Heaviside theta function in the denominator. The resulting curvature will be 0 everywhere except the pole $x = x_{0}$ where it diverges.
There is no physical example that I know of which exhibits any similar behaviour, but you can always find an infinite amount of metrics such as the one above, which lead to curvature tensors including delta functions in some of their components.
An important stipulation of course is that this kind of abrupt "jumps" and "bumps" on the metric are technically not well-defined mathematically when we perform differential geometry, since then the manifold you examine (your spacetime) is not smooth, thus a lot of the assumptions we take for granted in GR (General relativity) don't hold.
The above then remain relevant as a crude approximation of a metric that is smooth, but with abrupt changes at specific points.
A: Delta functions are discontinuous at the origin. A curvature that is exactly describable by a delta function isn't possible in spacetime as spacetime is seen to be continuous and smooth and hence curvature must be too.
As delta functions have a point that is infinite, the appropriate forms of spacetime, given the criticism above, are spacetimes with an infinite point of curvature.
Such spacetime are given by the Schwarzschild spacetimes that describe black holes and which have essential singularities where geodesics end. These are points of infinite curvature. This could well be one good reason for including torsionful gravity as they do not have singularities such as the Einstein-Cartan theory. It's worth noting that supergravities, which superstring theories reduce to in the weak energy limit, have torsion and so they too do not have singularities.
