Why did abiogenesis only happen once?

It may never have happened again, however, since all lifeforms on Earth today are similar on a molecular level (DNA), suggesting a common origin.

An important distinction to make here is that all extant life on Earth has a common origin.

It's completely possible that abiogenesis occurred many times, but whatever organisms emerged as a result became extinct early on or are not preserved in the fossil record.

Immediately after the first spontaneous abiogenesis, environmental conditions on Earth changed dramatically, making a repeat impossible.

Nope. The red bands and subsequent dating techniques suggest it took a very long time for a change to occur.

The first organisms that arose consumed any subsequent organisms...

Possible, but we have solid data suggesting the first functional organisms were autotrophs. That is, they made their own food from whatever they were exposed to, and the predator/prey relationship had yet to play a major role.

The common origin theory is in fact false, despite the similarities between organisms, and abiogenesis did occur multiple times in the same way.

The Common Origin Theory - as you've defined it - is almost certainly true. DNA is a constant, as is RNA, mitochondria, etc.

That, however, doesn't mean abiogenesis couldn't have happened more than once. As I said above, the Common Origin Theory applies to extant (living) species, not necessarily all organisms that have ever existed. Species go extinct all the time.

Spontaneous abiogenesis never occurred on Earth after all because the conditions never allowed it; instead, a proto-organism arrived on Earth from a planet where the conditions do (Panspermia).

That merely pushes the question of abiogensis back a bit. Why did it evolve on another world and not Earth, then? How did it survive in space? Why was it so well suited to Earth's environment if Earth-like planets are (relatively speaking) rare?

The standard theory of geological history is wrong (i.e. the Earth was in a "fertile" state for much longer before the eventual origin of life than commonly thought, due to some unknown mechanism causing radiometric dating to give wrong results).

Estimations and conclusions have come from more than radiometric dating. We have fossil records, geological records spanning continents, and even ice-core records from the poles that both confirm radiometric dating and establish themselves as independent evidence.

None of these seem likely to me, yet even less likely I find the idea that life arose spontaneously almost immediately (within a few hundred million years, possibly even faster) after the Earth's crust solidified, and never again afterwards.

Why?

What are the currently held theories on that matter? Both random speculations and references to relevant publications are welcome.

This SE is not the place for random speculations. I don't have publications handy, so I may return and edit my answer when I have the impetus.

The most widely held theory is that all extant life, with a few possible exceptions, has evolved from a single organism that existed billions of years ago. That organism may or may not have been the only one to arise. If it was one among many variations of progenitors, we have yet to find the progenitors in the fossil or biological records. Given the extremely long time span between now and the beginnings of life, we don't expect fossil records to exist of all the organisms that existed some 3 B.Y.A. so the best we can do is speculate on what might have existed to account for what we see today.

Some people think that abiogenesis did occur more than once, and that the results of those occurrences are still with us. Paul Davies, for instance, is a proponent of what he calls 'the shadow biosphere' (here's a link to a popular article on his views: http://www.theguardian.com/science/2013/apr/14/shadow-biosphere-alien-life-on-earth).

I've seen him talk about it, and while he has pretty much zero evidence for the idea, it's an intriguing thought. His argument is basically that we're not finding the organisms in the shadow biosphere because we're just not looking in the right places or in the right way; our discovery techniques are all biased towards our particular RNA/DNA-based life.

Note that this doesn't really put common origin in the ground, it would just mean that there's more than one common origin for each 'type' of life.

Also, it isn't necessary for early organisms to consume each other in order for there to have been multiple origins that subsequently went extinct. Our brand of life could simply have out-replicated the competition.

It may never have happened again, however, since all lifeforms on Earth today are similar on a molecular level

Actually, the similarity is at the chemical level where the chirality of all known-life's amino acids are "left-handed" (see http://en.wikipedia.org/wiki/Hypothetical_types_of_biochemistry#Alternative-chirality_biomolecules). Since there is no advantage to having left-handed proteins over right handed, we would expect the chemistry in a spontaneously organism to randomly chose one or the other. So, it's 99% likely that life didn't arise more then 7 times: (1/2)^7 < .01.

Or, all the descendants of the right-handed protein life could have died-off, but this seems unlikely if there is no inherent efficiency difference.

The first organisms that arose consumed any subsequent organisms that came into being. This was suggested by Alexander Oparin, but I find it very difficult to believe since it would have to happen everywhere on Earth

This is where your missing a big point:

• It's not necessarily eating them, or even outcompeting them once they arise, it's about cutting the legs out of the environment - eating the primordial soup of any loose complex organic matter. The places where the minimal materials for self replication machinery would have been available (since crystal-based origin of life was rejected in the 60's) would be very rare, and only on very specfic parts of the Earth...
• Two of the current theories for this location, were in seafloor thermal vents, or on beaches with radioactive sand. Once life arise once in one of these niches, it will gain a strong foothold in these exact locations, crowding out those places orginal "ideal conditions" that led to the generation of life in the first place.

Ultimately, we'll probably be better able to answer this question by looking out to other planets than by un-earthing any satisfying evidence that could be found intact from 3 billion years ago.

None of these seem likely to me, yet even less likely I find the idea that life arose spontaneously almost immediately (within a few hundred million years, possibly even faster) after the Earth's crust solidified, and never again afterwards.

You said it was okay to speculate, so here goes -

Note that we really are in a sweet spot where we've evolved intelligence and started thinking about expanding beyond our planet when a relatively short time is left (a few hundred million years) before the Earth becomes too hot for critters like us (when that does happen, it probably won't matter for us - I think since we would have superseded our Biology and will be able to thrive in almost any environment, but that's tangential). So, if the whole process had started a few million years late, we wouldn't be around having this cordial conversation. My suspicion is that this tragic circumstance plagues other planets where life might evolve and we've won the lottery in terms of life starting here soon enough. But then again, if we hadn't won the lottery, we wouldn't even be aware of it.

There are essentially two ways this can be interpreted:

1. Why did only one type of life evolve at the beginning, as opposed to multiple types coexisting?
2. Why have no other forms of life evolved much later than the one we know about (i.e. when relatively "modern" cells already existed?

I think the answer to 1) is "we don't actually know if it didn't". During the "RNA world" and earlier, it's quite possible that many self-replicating molecular machines evolved totally independently from each other and coexisted much like separate species today. However, once whole cells came along, any "free agent" self-replicating molecules that hadn't joined forces with others to make cells (or didn't evolve to exploit cells, like viruses) were likely left in the dust.

As for the answer to 2), similarly to the last comment above, once the familiar type of life was on a solid footing, it would have probably eaten or starved any competing form of life before it could get off to any kind of start. A crude new self-replicating molecule could never outgrow a highly evolved cell containing multiple already highly evolved self-replicating molecules within it.