Can black holes explode?

I don't have any link but "The first three minutes" by Steven Weinberg might be a good starting point.
Tullio

Firstly : there isn't yet a theory that deals adequately with the centre of a black hole or whatever existed at the instant of creation of the universe.

The Big Bang theory is a descriptive timeline that begins an instant after 'creation', indeed The First Three Minutes is an excellent summary of what observational evidence we have that leads us to think of that as actually occurring, compared to other propositions like the Steady State idea. Written in the 1970's by Steven Weinberg ( a subsequent Nobel winner in the area of particle physics ) it begins about 10^(-35) of a second from a point in time that, if we keep running the projector backwards, we would label as a point of infinities ( mass, density, energies etc ). Cosmic Inflation talks of even earlier fragments of time ( all this discussion assumes a meaning to the concept of time 'then' ). The CMB observations describes a surface of light scattering formed some 300,000 years later. Detail aside, the summary is : the universe is expanding, it was hotter and denser before than it is now, and we can't track using evidence available today what occurred beyond about 14 billion years ago ( give or take ). It's not a failed theory, simply a work in progress, as indeed all science is. Claims of it's demise are typically of the 'straw man' variety, ie. deliberate misrepresentations of the scientific discourse on the subject. After all : if you come into shop, the signage of which says 'Pizza' out front, then the lack of motorbike repair facilities inside ought be no surprise.

As for scientific method generally, most scientists I'm aware of don't require 'repeatability' but 'reproducibility'. There's an important difference. Nothing is repeatable in the strict sense. One can't arrange the universe, Big Bangs included, to be in any exact state ( if indeed the universe has a quality called 'exactness', see quantum mechanics ) for a re-run of results. However, for the variables declared to be of interest in some arrangement one can purport ( as a declaration of the standard of acceptable evidence ) that results be within some error tolerance if the same technique and setup are used elsewhere in time and space. Again, critiques of scientific method are all too keen to create 'straw men', by stripping experimental practice and observation of a myriad of well known pragmatic measures. So you wind up with people saying that Newton's theory of gravity is 'only a law', but unusually without provision of contrary examples in line with their beliefs, like them jumping off a cliff say ...... I would say "of course it is 'only a law'", but don't miss the practical point that you will die if you're not careful near long drops. Gravity attracts regardless and science is a practical creed. I generally leave such people alone with their thoughts, and specifically don't get into cars with them.

Now the word 'singularity' is worthy of discussion. In a strict technical sense it can be a matter of how one chooses to describe something.

Suppose I have a row of soldiers on a parade ground, all in uniform and in a neat line. This is a view from 'on high' meaning that I have a complete sense of the arrangement. Pretend that I am limited now in what I can measure : I have a long straight stick ( as long as the line of soldiers would be nice, but not required ) that I press against the chests of the soldiers. So I'm seeing them front on. Each soldier I can mark the position of, literally, using the stick. Each soldier has his own mark, not equal in position along the stick to any other soldier's mark. Now go to one or other end of the line and look along it. The soldiers now appear super-posed by eye. My stick, which I hold against the nearest soldier's arm ( as what would seem to be at a right angle to the line of soldiers when viewed from above ) cannot in this circumstance separate the soldiers. They will all share the same mark on the stick. This is the rough idea of a 'co-ordinate singularity' : a particular choice of measurement method has lead to ambiguity or overlap.

The event horizon of a black hole is a co-ordinate singularity. When seen from afar everything on it appears pasted onto a 2D surface. So the third dimension ( measured radially out from the hole's centre ) is the co-ordinate upon which stuff overlies, like the soldiers. However if one takes the viewpoint of a traveller descending inwards then nothing especial happens as the radius ( seen from afar as the event horizon ) is passed - well nothing happens on account of co-ordinate choice at least. Black holes with a relatively small radius have a fierce gravity gradient ( compare with the acceleration due to gravity being pretty constant near the Earth's surface ) that could rip you apart. Rather large black holes, with many millions of Sun's worth inside them, have large radii and the gradient at the radius of event-horizon crossing is quite mild.

The difficult singularities are those which can't be 'transformed away' by a suitable choice of co-ordinate system. The centre of a black hole is such. All choices give infinities. Ditto for the time T=0 in the Big Bang scenario.

Note that General Relativity does not prescribe what sort of universe we live in. It 'merely' determines how one state moves to the next, or how a situation over here will affect something over there. You need what are called 'initial' or 'boundary' conditions. For instance the modelling of a black hole assumes that far way from it ( at 'infinity' ) spacetime is flat. Or if you like : that the black hole is alone in the universe. Approximations. You can then compare what things appear to be from different perspectives : the far observer looking at the descending matter ever gradually going splat on the event horizon, or along for a ride with the descending matter itself heading for oblivion further in. As for the Big Bang, while gravity is attractive for sure, that doesn't exclude the effect of outwardly directed pressure or kinetic energy and whatnot. One can throw a stone upwards, and if hard enough it will orbit the Earth and perhaps leave us entirely. But there is much debate and little settled on these ( pre ) conditions for the early universe. Just beware overstatements. Stick to observables.

One line of thinking upon the Big Bang starts with the question 'where did it happen?' The answer is 'everywhere'. So what our now labelled as separate physical points some 14 billion years later, were once identical. Thus the Big Bang at T=0 is a co-ordinate singularity, but much more besides. Looking backwards in time the ruler collapses, very like the coiled-spring-retractable-metal-strip-with-a-button type builders use ( and as a child I would play with until it's destruction or confiscation by Dad ).

Cheers, Mike.

( edit ) That is : until I destroyed it, and before Dad could take it from me .... :-)