The young centre of the Earth (2016)

Tests of gravitational time dilation are also within the reach range of enthusiastic amateurs using second-hand atomic clocks:

Part 4 (starting pdf page 44) of http://web.stanford.edu/group/scpnt/pnt/PNT18/presentation_f... (slides)

Or if you prefer non-PDF:

http://leapsecond.com/great2005/

http://leapsecond.com/great2016a/

Yeah: "The two measurements consist of approximately 100,000 s of low-height data and 40,000 s of high-height data, and the clocks exhibit (Fig. 3) a fractional frequency change of (4.1 ± 1.6) × 10^−17."

If there are any such stable deep-crust locations, they're most likely located under a continental location rather than oceanic, if my understanding is correct. Ocean basins reform at a higher rate (and hence: are more-recently-exposed crust) than continental plates.

The very oldest regions are two locations, presently in South Africa and Australia, previously joined, dating back several billions of years.

The Jack Hills region in Australia has been dated to 4.4 billions of years (the Earth itself is 4.5 billion):

https://www.nationalgeographic.com/news/2014/2/140224-oldest...

A region presently under Greenland was dated to 3.8 billion years, in 2007:

https://www.newscientist.com/article/dn11438-oldest-chunk-of...

The Nuvvuagittuq Greenstone Belt, near Hudson Bay, Canada, has been dated to between 3.7 and 4.3 billion years old:

https://en.wikipedia.org/wiki/Nuvvuagittuq_Greenstone_Belt

All of these are surface rock. My thought is that drilling from these locations might find very old subsurface structures as well. Though whether any relativistic time dilation could be observed is hard to say.

Not disagreeing that there's a point of no return wherein an infaller will inevitably collide with the singularity, just whether it's really correct to say that makes the singularity a moment of time. For starters, different infallers can arrive at the singularity at different times (even ones that arrive at the horizon at the same time can be relatively accelerated, resulting in having different interval lengths).

Also not sure it makes sense for non-eternal BHs. For a BH that forms by gravitational collapse and eventually completely evaporates, surely the singularity has a timelike worldline? Some "lucky" too-early infallers (a cosmic neutrino, say) might pass through the centre collapsing progenitor before the horizon forms, while too-late infallers might pass through the region after final evaporation (assuming no horizon-equipped remnant). The "just-miss" is depends on where in spacetime the coincidence happens, and any reasonable slicing or threading would attribute the miss to being at the wrong moment in time.

Maybe easiest to think about that if the singularity is not always at the spatial origin of a system of coordinates. For instance, if SN1987a's remnant contains a black hole, that black hole is surely moving around the galaxy with the luminous matter "now", but before the final collapse there was neither horizon nor singularity.

I'm also perhaps unreasonably superstitious about the slogan's survival of arbitrary parameterizations, e.g., one can do an affine parameter on even an eternal BH's singularity and have what looks like a decent proper time for it.

Perhaps another way of putting it is that the usual Carter-Penrose BH diagrams showing long horizontal wavy line segments for the singularity are misleading because of exaggerations caused by the particular conformal chart it uses, kinda like how the Earth's very-near-polar regions are not that big even though they look that way on Mercator charts (those are also conformal, and you get really different results in the near-polar regions in other conformal projections even as closely related as a transverse Mercator/Gauss conformal projection, https://en.wikipedia.org/wiki/Map_projection#/media/File:Usg... versus https://commons.wikimedia.org/wiki/File:Usgs_map_traverse_me... ).

Just thinking aloud in the wee hours, and maybe having given too little weight (pardon the pun) to the qualification "inside the horizon" just before your close paren. There's a lot of rigour that can be hidden in those three words.