Black hole singularities are created after the core of a very massive star collapses beyond an imaginary sphere called the event horizon. Since these singularities are infinitesimally small, they possess infinite density.
The differences are that black hole singularities, of course, do not contain all of the matter and energy in the Universe because there are so many of them. There was only one Big Bang singularity, and it contained the whole Universe. Another difference is that space and time were born from the Big Bang singularity, and black holes actually stretches out space to the point where it probably rips the fabric of space-time, and ends time altogether.
According to Stephen W. Hawking, singularities either occur entirely in the future or entirely in the past. In this case, black hole singularities always lie in the future (if you were to go near one, your time will come to an end), and the Big Bang singularity always lie in the past (if you were able to rewind time, galaxies will become closer together, eventually coming together at a point).
Answered by: Joel Novicio, Undergraduate Physics Student, South San Francisco
In both cases the singularity was a single point where the curvature of space time are infinite. It is believed that at this point the general theory of relativity (almost universally accepted as 'the' accurate description of gravity) ceases to hold true.
The singularity of a rotating black hole is essentially the same except that it exists in a ring (thus the name ring singularity). The consequences of a rotating black hole (if they exist - we have no direct proof that any type of black holes exist) however, are very different from non-rotating ones.
Answered by: K Shaban, CS/Physics Student, Carnegie Mellon
The big bang singularity is where all the mass of the universe used to be concentrated. It had all of the properties of a black hole singularity but from it 'grew' space time and matter was released into this space as the fundamental particles of very high energy. This is the big bang. Therefore, the main difference is that a black hole singularity is the end of space time (and pulls matter in) and the big bang singularity is the beginning of space time (where matter and space were made 'real').
Answered by: Martin Archer, Physics A Level Student, Surrey, UK