Well, first:  Does any collapse theory have any experimental support?  No.

With that out of the way...

If collapse actually worked the way its adherents say it does, it would be:

1. The only non-linear evolution in all of quantum mechanics.
2. The only non-unitary evolution in all of quantum mechanics.
3. The only non-differentiable (in fact, discontinuous) phenomenon in all of quantum mechanics.
4. The only phenomenon in all of quantum mechanics that is non-local in the configuration space.
5. The only phenomenon in all of physics that violates CPT symmetry.
6. The only phenomenon in all of physics that violates Liouville's Theorem (has a many-to-one mapping from initial conditions to outcomes).
7. The only phenomenon in all of physics that is acausal / non-deterministic / inherently random.
8. The only phenomenon in all of physics that is non-local in spacetime and propagates an influence faster than light.

WHAT DOES THE GOD-DAMNED COLLAPSE POSTULATE HAVE TO DO FOR PHYSICISTS TO REJECT IT?  KILL A GOD-DAMNED PUPPY?

- Eliezer Yudkowsky, Collapse Postulates

In the olden days of physics, circa 1900, many prominent physicists believed in a substance known as aether. The principle was simple: Maxwell's equations of electromagnetism had shown that light was a wave, and light followed many of the same equations as sound waves and water waves. However, every other kind of wave- sound waves, water waves, waves in springs- needs some sort of medium for its transmission. A "wave" is not really a physical object; it is just a disturbance of some other substance. For instance, if you throw a rock into a pond, you cannot pluck the waves out of the pond and take them home with you in your backpack, because the "waves" are just peaks and troughs in the puddle of water (the medium). Hence, there should be some sort of medium for light waves, and the physicists named this medium "aether".

However, difficulties soon developed. If you have a jar, you can pump the air out of the jar, and then the jar will no longer transmit sound, demonstrating that the wave medium (the air) has been removed. But, there was no way to remove the aether from a jar; no matter what the experimentalists did, you could still shine light through it. There was, in fact, no way of detecting, altering, or experimenting with aether at all. Physicists knew that aether must be unlike all other matter, because it could apparently pass through closed containers made of any substance. And finally, the Michelson-Morely experiment showed that the "aether" was always stationary relative to Earth, even though the Earth changed direction every six months as it moved about in its orbit! Shortly thereafter, the inconsistencies were resolved with Albert Einstein's Theory of Special Relativity, and everyone realized that aether was imaginary.

Shortly thereafter, during the 20th century, physicists discovered two new forces of nature: the strong nuclear force and the weak nuclear force. These two forces, as well as electromagnetism, could be described very well on the quantum level: they were created by the influence of mediator particles called (respectively) gluons, W and Z bosons, and photons, and these particles obeyed the laws of quantum mechanics just like electrons and mesons did. The description of these three forces, as well as the particles they act upon, has been neatly unified in a theory of physics known as the Standard Model, which has been our best known description of the universe for thirty years now.

However, gravity is not a part of this model. Making an analogy to the other forces, physicists have proposed a mediator particle known as the "graviton". The graviton is thought to be similar to the photon, the gluon, and the W and Z bosons, except that it is massless and has spin 2. I posit that the "graviton" is essentially the same theory as the "aether": a misguided attempt to explain something by reference to similar-seeming things that were explained in the same way. Consider the following facts:

• Some of the brightest minds in the world have been working non-stop on how to to extend the Standard Model/quantum mechanics theory of forces to gravity, for more than forty years, and we have yet to find a satisfying way of doing it. Theories of physics have a known tendency towards elegance and simplicity, and nothing we have come up with in the past forty years is either simple or elegant, let alone both.
• Of the explanations that have been proposed, string theory is currently the most popular. The idea that a theory of physics can have "popularity" should, in and of itself, be a warning sign. There can only ever be one reality, and so there can only ever be one correct theory of reality. The physics community, of course, cannot all just switch to a new theory as soon as its correctness has been proven; there are lots of inevitable time lags. However, we have had ongoing disputes between people in the string theory camp, people in the loop quantum gravity camp, and people in other camps for more than thirty years now, with zero sign of a winner or of the matter resolving itself anytime soon. This, to me anyway, is a sign that no one really knows what is going on, and lots of people are just following their monkey intuitions and forming Blue and Green teams.
• String theory, of course, has zero testable predictions. Loop quantum gravity, another graviton-based theory, also has zero testable predictions. In fact, to my knowledge, all of the use-gravitons-to-explain-gravity-on-a-quantum-scale theories have zero testable predictions.
• String theory has additional problems: it's not a satisfying explanation in the same way that relativity, or Newtonian gravity, or Keplerian astronomy, or Maxwell's laws are, because there are so many different versions of it. You can't just go read a textbook on string theory, and say "Aha! So that's how the world really works", because there are a ton of different kinds of string theory. There are (as of this writing) bosonic string theory, type I string theory, type IIA string theory, type IIB string theory, and heterotic string theory. (This list does not include the different ways of describing the universe within each of these theories, which are practically infinite.) It also requires numerous additional dimensions, which are unobservable and untestable, but which are required to make the math work right.

And, with reference to the graviton itself:

• Gravitons have, of course, never been detected, and (so far as I know) there aren't even any serious proposals on the table for how to detect them. In fact, it is widely thought that gravitons have such low interaction cross-sections that detecting them with any physically possible detector is a hopeless proposition.
• The classical theory of gravity that we know works well, General Relativity, makes no mention whatsoever of gravitons, and describes gravity in terms of the curvature of space-time. None of the other three forces (weak, strong, electromagnetic) have ever been described in this way, but gravity has, with a precision of more than twelve decimal places, in addition to many other unique predictions (such as the geodetic effect, which has recently been confirmed experimentally).
• Gravity is much, much, much weaker than all of the other forces. In fact, it is more than thirty-five orders of magnitude weaker than both the electromagnetic force and the strong nuclear force. 
• Gravity is nonrenormalizable, which means that, if you try to calculate the strength of the gravitational interactions using a quantum-mechanical, quantized-graviton model, there's no simple way to make infinities go away, like there is for electromagnetism, the strong force and the weak force.

So, what's really going on here? I don't know. I'm not Albert Einstein. But I suspect it will take someone like him- someone brilliant, very good at physics, yet largely outside the academic system- to resolve this mess, and tell us what's really happening.