Dear Antony,
Now let me explain in detail for the benefit of clarity. As a description from OUTSIDE this system possibly yes we could call this situation a singularity. But one could also describe this situation from WITHIN the system (and which I think is the more useful approach as a first principle). Then this situation will qualify physically as a LENGTH SCALE (think, "radiation gauge", "renormalization").
We could even take a comparative view of this situation so it means the SUPERPOSITION or, perhaps most generally, the CONSERVATION LAW.
In a wave model of the situation, the point is that we may think of any observer as simply the phase-space (say, "wave speed" of Huygens' principle or "constant" of Snell's law). And finally, in material terms this unobservability/observer should be the "matter wave" (wave function).
And sure enough there is phase-space formulation of QM, by Weyl, Wigner, Groenewold, Moyal etc.
But I would rather that we go all the way with this approach by assuming any OBSERVABLES (i.e. a "locality" or "position" notion) as simply the phase-points or harmonics vis-à-vis any OBSERVER as the phase-space (length scale, non-locality, invariance).
And by this we will be talking then of any relevant "observer" as the wave nature proper (as against his observables/harmonics as the corpuscular nature) or indeed vice versa.
Thus the concept of a pilot wave emerges naturally as defining the maximum wavelength i.e. the "fundamental frequency" or "period". It is sort of like the probability unity as equals the invariance (think, "energy") so a "probability amplitude" is simply a dependent variable/observable/harmonic (think, the "forms" of energy).
Forgive my lengthy talk. But I hope this makes me clearer.
I find our essays share some basics.