(this answer addresses the new question)
As a consequence of the singularity theorems, it is not only possible but (arguably) inevitable for singularities to form in a finite amount of "time" in a physically reasonable spacetime. The word "time" in this context means "proper time along a specific timelike geodesic". For example, if there is a trapped surface* in spacetime, then a singularity will appear within a finite amount of proper time (along a timelike geodesic) in the future of that surface; so, an observer sitting in a collapsing star will reach the singularity in finite time. Thus, the collapse of matter is one possible way to create a singularity "out of nothing". If your spacetime is globally hyperbolic and you foliate it by Cauchy surfaces you can say in a much more "universal" way that the singularity didn't exist at time [;t_0;] and came to exist at time [;t_1;].
I should point out that the singularities are a generic feature of physically reasonable spacetimes; take a look at the Hawking-Penrose theorem - it applies in very general situations.
Also, as the original question was about black holes and not singularities, I should advise you to make a clear distinction between the two concepts. Trapped surfaces form due to the condensation of matter (this is the famous Schoen-Yau theorem), and under a certain extra hypothesis, these surfaces will be hidden inside black holes. This extra hypothesis is the well-known (weak) Cosmic Censorship Conjecture (CCC). If it does not hold, gravitational collapse can create naked singularities, that is, singularities not "causally hidden" by the event horizon of a black hole. Much of what is known in general about black holes depend crucially on the CCC.
*A trapped surface is a two-dimensional spacelike compact surface such that the null geodesics departing from it are accelerating towards each other - mathematically, we say that the expansion of the congruence of future-directed null geodesics orthogonal to the surface is negative.