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It should be possible for private citizens (well maybe companies due
to the cost) to detect underground caverns with ground penetrating
radar. The low frequency kind goes deep, but is lower resolution.
Looking around the net, I see they can go as deep as 80 meters using
16Mhz. I found a reference to GPR at 10Mhz, but no depth
specification. A plan might be to get some magazine to sponsor the
analysis.

An alternative way to find caverns is by generating an impulse and
monitoring the propagation speed at various locations. The void area
will not propagate the impulse. This is one technique used in oil
exploration, though there the goal is to find a change in density, not
an empty cavern.

Folklore is useful in knowing where to investigate, but nothing beats
physical evidence.