How can one describe this phenomena? What are the factors that we should consider for solving such a problem?
I want to extend PM 2Ring's comment to the other answer because I think it is important.
For a Schwarzschild – non-spinning – black hole it is critically important to realise that the singularity does not exist yet. What that means is that the singularity is not in the past for anyone or, to be more precise, there are no timelike or null curves which have the singularity in their pasts. This is true both outside and inside the horizon: you can't observe the singularity from inside the horizon because it is still not in your past so no information can get from it to you.
The difference between being outside the horizon and inside the horizon is that, if you are inside the horizon, the singularity is all of your future, while if you are outside it it is only some of it. In other words, from any event outside the horizon, not all future-directed timelike or null curves meet the singularity, while from any event within the horizon all of them do.
So, quite independently of whether a singularity – something which has to be removed from the spacetime manifold – 'exists', and the problem that GR is almost certainly not a correct theory in the extreme conditions close to a singularity, I think it is very reasonable to say that no, the singularity doesn't exist. GR says it will exist for observers which pass through the horizon, but GR is almost certainly wrong about that.
But Schwarzschild black holes are not the only kind (and probably never actually occur in practice, since there will always be some angular momentum). So a more general question is: can singularities exist (be in the past light cones of observers) for other kinds of black hole? This question is normally framed as the cosmic censorship hypothesis, due originally to Roger Penrose. To my knowledge, which may be out of date the cosmic censorship hypothesis is still an open question.
[In terms of classical GR there's one big exception to this: the singularity at the big bang is in everyone's past and thus in principle is observable by everyone.]
The question is really about how to turn a loose question into something that has an answer; to some degree this is a logic and philosophy question rather than astronomy, so don't get surprised if it gets closed.
The key thing here is the concepts "really exists" and "with respect to relative frame of reference" - what does that mean?
In relativity theory there are many possible frames of reference, typically described that what different kinds of observers would observe or as different coordinate systems for the same underlying spacetime. The theory describes how to translate from one frame to another one.
Can something really exist just in relation to some frames of reference? In pure special relativity events and objects have existence independent of the observers, so there the answer is no. There are still things like simultaneity that are bound to particular frames and do not translate, and there are events that cannot be seen from other events due to lightspeed limitations.
In general relativity frames can be curved in complicated ways and whole parts of spacetime can be invisible to some frames. An obvious example is black hole interiors: events there cannot signal to observers located outside. Choosing other coordinate systems can allow reasoning about the whole spacetime, but most do not correspond to anything an actual physical observer could ever observe: they are frames of reference for idealized mathematical observers.
So, does the singularity "really exist" in a frame of reference? Yes, mathematically it is not a problem to extend the spacetime manifold towards the $r=0$ singularity. Note that you cannot include it in the spacetime manifold: it is in a sense the edge, just as the number $1$ is not part of the set of numbers $<1$ despite there being numbers coming arbitrarily close. But it makes sense to speak of a region in spacetime where curvature diverges to infinity as one approaches the edge, so most people slightly sloppily talks about the singularity as being inside whatever reference frame is being used. Mathematicians will disagree.
Note that this is more philosophy of relativity theory issue than a discussion of what is actually going on in actual black holes. Answering this kind of question requires formalizing the words into terms of the theory we use, and then checking the revised question against the theory.