At the outset, let me  begin with some preliminaries: Consider an evolving spherically symmetrical fluid and divide it into concentric spheres in such a manner that, despite evolution, the number of particles (or baryons) remain fixed for each sphere.  This fixed number for a given sphere could be proportional to a symbol  `r' and which can be a  `co-moving' radial  coordinate. At the surface of such a sphere, let there be a clock tied to the fluid. The time recorded by such a tagged clock ``t'' is co-moving time. The surface area of the sphere, in curved space-time, need not be proportional to r2; on the other hand, let this changing surface area be 4 pi R2 (r,t). This ``R'' is called the area coordinate.

Let the gravitational mass within a certain sphere be M(r,t). In Newtonian gravity, even if there would be emission of radiation during collapse, M(r,t) would remain unchanged because there is no E=Mc2 law there. Thus the value of Rs = 2GM/c2, the  sort of instantaneous Schwarzschild  radius is fixed in Newtonian gravity. Hence as the star would shrink, it is plausible  that any section of it having `radius'  R(t), will plunge beneath the its Rs:

R(t) < Rs = 2G M/c2  or

2GM/R c2 >1

In fact, this is the basic concept of a `Trapped Surface'.

But in General Relativity (GR), emission of radiation means loss of mass. In fact, whether it is Newtonian or GR case, realistic gravitational collapse must be accompanied by emission of radiation:

Thus, in GR, one cannot guarantee for formation of trapped surfaces unlike the Newtonian gravity. However, in 1965, Sir Roger Penrose,  insisted that, for continued collapse, sooner than later, trapped surfaces must form, i.e., and any section of the fluid must dip below its running Schwarzschild surface:

R(r,t) < 2GM(r,t)/c2 or 2GM(r,t)/Rc2 >1

Later, Hawking&Penrose extended this assumption to all situations, and for non-spherical cases too. Once the assumption of such a deadly one-way trap door will be made, even a pulse of light emitted outward would tip inside and fall inward as it happens inside the Event Horizon of a black hole (EH)! When such a drastic situation will be assumed to occur, naturally, with the help of some other reasonable assumptions, formation /existence of space-time singularity for continued collapse, as well as for wide range of cosmological scenarios
would be inevitable. However way back in 1972, Misner, Thorne&Wheeler commented in their Classic Monograph Gravitation that the assumption of formation of trapped surfaces hardly look to be reasonable.
In fact, in 1990, Senovilla offered an explicit example (no general proof) of non-occurrence of trapped surfaces for a cosmological model having Cylindrical Symmetry:

I was however unaware of such development and believed in the inevitability of singularities and black holes like all other astrophysicists. But by 1997, by noting that it was difficult to understand emission of mind-boggling amounts of energy from the Cosmic Gamma Ray Bursts, I wondered whether trapped surfaces were ever formed. In 1998, I had series of discussions with one noted relativist from TIFR and he found no problem (at-least at that time) with my proof.
More importantly, I also had detail correspondence with Prof. P.C. Vaidya, the most respected and famous Indian GR expert. He  told me that I was on correct tract and my work looked very attractive from physical perspective:

My twin papers received modest comments in PRL and they were rejected. When I met Prof. Vaidya following this disappointment, he suggested  that he would forward my two manuscripts to ``Current Science'', a respectable journal from the Indian Academy of Sciences. With his recommendations they were likely to be published there. However, I decided that, I would publish my proof independently, and submitted a highly extended long manuscript to Foundations of Phys. Letters (FPL). But for months, the editor wound not respond to my queries. Then I noticed that one of the Editorial Board members of FP was Prof. G. 't Hooft, and I requested him to ensure that my manuscript got duly processed. It is only after his prodding that the editor replied back: He had not been able to find a suitable referee and  planning to return my manuscript. However now he worked overtime for its processing; and it got accepted for publication in 2000 only after 2 anonymous referees consented.
Though this long paper discussed many aspects of black holes, the central proof did not involve any definition of fluid velocity; on the other hand, it only exerted that a TIME-LIKE world-line of the fluid element must always remain time-like. And since my proof showed that, any formation of trapped surfaces would lead to the violation of this cardinal rule,  it was concluded that there must not be any trapped surface.

Following me, Robertson&Leiter too claimed that (even) for spherical collapse, trapped surfaces must not form in order to protect the cardinal GR rule: `Once Time-like, Always Time-like':

D. Leiter&S.L. Robertson, Foundations of Physics Letters for February 2003 ;arXiv:astro-ph/0111421v3
Later I found that my proof left some room for confusion on a subtle aspect, and accordingly, I offered a transparent proof in 2004:

In view of the importance of the theorem, I kept on  fine tuning it:

Some staunch BH believers however may try to cast aspersion on such exact proofs by a vicious
circular logic:

1. First they will insist that trapped surfaces must be there as assumed by the singularity theorems. 

2. And since the world-line of a fluid element must be  time-like, they will argue that, even if trapped surfaces would form during collapse,  the fluid world-lines must remain time-like. Therefore, they may argue that, any proof which shows the contrary must be incorrect. And eventually, they may insist that since the no-trapped surface formation proof is incorrect, trapped surfaces must form!

And it is only this year that I came across another previous proof (entirely different from mine) by Kriele which is applicable for only a uniform density spherical fluid:

Non- formation of trapped surfaces instantly rules out formation of finite mass black holes. It also puts a question mark on the physical reality behind the  mathematical claims of ``Naked Singularity'' formation.

 Now one will wonder then ``What about the EXACT Oppenheimer - Snyder solution about the formation of black holes?''

Well, OS assumed the fluid to be a non-radiating pressure-less ``Dust''. It transpired that pressure can  exactly vanish if density will vanish too. Thus, in reality, a dust has no gravitational mass, and this was proved from Birkhoff's Theorem. When M=0, there is no physical gravitational collapse associated with a dust, and there is no BH. Thus the famous OS solution is only a mathematical illusion:

Thus, in reality,  there is no exact   general relativistic  collapse solution for the formation of either any `trapped surface'' or any finite mass `black hole'. This statement must be distinguished from the fact, on the other hand, there are exact solutions like the Schwarzschild and Kerr solutions   and which apparently  indicate the  existence of true black holes. In fact, in GR, there are more than 4000 exact solutions, and it is already known that most of these solutions do not represent any real astrophysical objects or models  in the physical universe. Similarly, even the black hole exact solutions need not represent real astrophysical objects which may form by gravitational collapse. One can resolve this puzzle by realizing that the exact solutions involve various integration constants whose values could  actually be zero.