The Schwarzschild metric is given by,
$$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}t^2 -\left( 1-\frac{2GM}{r}\right)^{-1}\mathrm{d}r^2 -\underbrace{r^2\mathrm{d}\theta^2 -r^2 \sin^2 \theta \mathrm{d}\phi^2}_{r^2\mathrm{d}\Omega^2}$$
in spherical coordinates. Notice the metric tensor is singular at the origin, $r=0$, and at the Schwarzschild radius $r=2GM$. The former is a true physical curvature singularity. However, as one can verify by computing various curvature scalars, the singularity $r=2GM$ is unphysical, and can be removed by a coordinate transformation. In particular, the metric in Kruskal coordinates is given by,
$$\mathrm{d}s^2 = \frac{32G^3M^3}{r}e^{-r/2GM} \left( \mathrm{d}U^2 -\mathrm{d}V^2\right) + r^2 \mathrm{d}\Omega^2$$
The event horizon $r=2GM$ (in natural units) corresponds to $V=\pm U$, and indeed the metric is not singular at the point, reflecting the fact the singularity arose simply because we chose an inappropriate coordinate system. Regarding the $r=0$ singularity, we cannot define a notion of length or associated any length to the singularity; it is a single point on the manifold.
At the Planck scale, we cannot resort solely to general relativity, and quantum gravity effects become important. From a quantum field theory perspective, this is typical. Short distances correspond to a high energy scale where our theory does not provide a description. As Prof. Tong states,
The question of spacetime singularities is morally equivalent to that of high-energy scattering. Both probe the ultra-violet nature of gravity. A spacetime geometry is made of a coherent collection of gravitons. The short distance structure of spacetime is governed - after Fourier transform - by high momentum gravitons. Understanding spacetime singularities and high-energy scattering are different sides of the same coin.
Nevertheless, gravity is different from simply an effective field theory, such as Fermi's theory of the weak interaction. We may still make predictions regarding gravity for high energies; e.g. if we collide at energies above the Planck mass, we know we form a black hole.
A final remark as other answers mentioned naked singularities. When quantum gravity effects become important, after we pass the event horizon, we cannot communicate them to others behind the horizon; this is an example of cosmic censorship. However, violations have been hypothesized, c.f. the Gregory-Laflamme instability of $p$-branes and black strings.
Best Answer
In Thorne's book, on page 477, it says "Because all conceivable curvatures and topologies are permitted in side the singularity, no matter how wild, one says that the singularity is made from a probabilistic foam. John Wheeler, who first argued that this must be the nature of space when the laws of quantum gravity hold sway, has called it quantum foam."
So what you are thinking of as a foam is really only a probabilistic foam. This is just a way of saying that every possible state in the singularity is only a probability. (Takes you back to the probabilistic nature of any particle which is a discussion far beyond this answer).
But remember Thorne's central point of the chapter, that time does not exist at the singularity. Space and time have separated from each other. Time stops at the event horizon, but space continues to dilate all the way down to the singularity. This quantum foam is Wheeler's way of describing the situation of an unknowable, but only probabilistic, nature of the singularity.