How far away is the event horizon of a (Schwarzschild) black hole away from the central singularity for a radially infalling observer starting with $v=0$ somewhere outside the black hole? After crossing the event horizon, such an observer hits the singularity in a finite time, hence such an observer would also assign a finite distance from the horizon to the singularity.
"Crossing the horizon" shall mean that the observer moves from outside the black hole (there are future world-lines, including non-radial and non-freefalling ones, that do /not/ hit the singularity) to inside the black hole (all future world lines hit the singularity).
The radius of a black hole is defined as follows: Take a ball $B$ in flat (Euclidean) space that has the same surface area like the event horizon of the black hole. Then the Schwarzschild radius of the black hole is defined to be the radius of $B$.
I'd guess that the so defined Schwarzschild radius is not the same (smaller?) like the proper distance from the event horizon to the center, but what is the ratio of these two values exactly, for example in terms of the mass $M$ of the black hole?
[EDIT]: Clarified that it's for a free falling observer.
Best Answer
You refer to the "central singularity," but the singularity of a Schwarzschild black hole is not a point at the center of the event horizon. It's a spacelike surface that is in the future of all observers. It's also not a point. See Is a black hole singularity a single point? .
The question you ask doesn't have a meaningful answer. From a point on the horizon, you can draw a null geodesic that intersects the singularity, and its metric length is zero. You can also draw a timelike geodesic, in which case the metric length will be (for +--- signature), a positive real number of order M in geometrized units. You can also draw a spacelike curve whose length in this metric is an imaginary number.
You refer to "proper distance," but that doesn't succeed in resolving this ambiguity. Proper distance is distance defined by a ruler at rest relative to the thing being measured. Inside the horizon, we can't have a ruler at rest. The spacetime inside the horizon is not static.