On the singularity $r=0$ of the Schwarzschild metric I faced following sentences:

Unlike the co-ordinate singularity at $r = 2M$, the origin of the Schwarzschild metric $r = 0$ has a true curvature singularity. It was first believed that this singularity was an artifact of spherical symmetry and that a generic collapse would evade the singularity. However, work by Hawking and Penrose showed that this was not so and that singularities were generic rather than special. The ubiquity of singularities is guaranteed by the singularity theorems by Hawking and Penrose.

Question1: What is the meaning of "generic collapse"?
Question2: When is a singularity an artifact of spherical symmetry?
 A: I believe that both questions are easily answered even without a GR background. Saying that something is an artifact of spherical symmetry means, in this context, that it was expected that the singularity would not occur in cases where no perfect symmetry is assumed. Since perfect spherical symmetry only ever occurs in theory, this would solve the problem of the singularity: it would never appear in practice. The real-life case is referred to as 'generic', denoting pretty much that it does not have any special properties (like perfect spherical symmetry) associated with it.
A: In addition to Danu's answer, it is worth mentioning that there is another class of solutions in GR which arise as the artifacts of special symmetry: naked sigularities (that is, singularities without event horizons).
The cosmic censorship hypothesis states that for a 'reasonable' matter, dynamical evolution from generic initial conditions can never produce a naked singularity. 
Nevertheless, there exist solutions in which the collapse of matter results in the formation of naked singularities. It's just the initial conditions for such solutions are 'special', often possessing high degree of symmetry (such as spherical symmetry), and thus would likely could not be realized in reality. 
Interestingly, the existence of such naked singularities was the subject of a bet between S. Hawking and J. Preskill/K. Thorne. Hawking conceded the original version of the bet "on technicalities" accepting that naked singularities can form under very special "nongeneric" conditions, and proposed a new version, which is still open.
