There's nothing special going on locally at the event horizon. And on the other hand, seed AI FOOM is just about the most singular kind of historical Singularity I can think of. The entire Hubble volume could undergo some kind of phase transition at the speed of light. (Maybe even faster.)

Not actually literally singular, mind you. Because it's 'real', not 'abstract'. But not even literally a 'co-ordinate singularity', figuratively speaking. (Unless Penrose is right about noncomputable physical action being involved in mental activity. [That is a joke. Perhaps I should label them. The 'hyperbolic' thing was a rather clever pun, by the way.])

So I'm not sure what you're getting at in your first sentence, which is possibly due to lack of sleep. Also:

if one starts with a differential equation that has a large rate of growth with respect to the function itself it isn't very hard to force a singularity.

I somehow have no idea what this means. But I would like to. Maybe it'll be clearer in the morning.