> I wonder: is a stock market bubble (the result of)
 > spontaneous order?
 > Daniele Besomi

It is not designed or commanded by a governmental authority,
but if the bubble is an outcome of subsidies that distort
decisions, then it is not truly spontaneous.
That is why in my judgment, the concept "spontaneous"
must apply not just to an order outcome but also to
the human action that created the outcome.

 > People quite rationally buy in the expectation of a price
 > increase,

But if that increase (such as in real estate) is due to
subsidies and governmental pressure,
then the outcome is not caused by a free market,
not really spontaneous.
See e.g. the recent book by Thomas Sowell,
The Housing Boom and Bust.

 > and price actually increases, confirming that they have
 > been (individually) rational. But at some point the bubble
 > bursts, for no rational reason whatsoever:

The bursting is rational, because the price of the asset
was pushed beyond a price based on the yield of the asset.
At the peak, speculators speculate on what other speculators think
rather than on the fundamentals of the asset.
Eventually, the asset becomes unafordable for those who seek
a true investment, and the price collapses.

 > the individually rational actions prove to
 > be collectively insane,

Why "insane"?
The speculators know or should know the risk.
Or they expect government to bail them out.

 > is it a kind of order?

It is an order in that there is coordination among individual choices.
The spontaneous order does not preclude choices we would consider odd 
or excessively risky.

Fred Foldvary