What defines which universe I will end up in? Following this question about the Many Worlds Interpretation of QM in which MW is stated to be deterministic: What "chooses" which universe I will be in, i.e., which outcome I will see?
I actually think that this is random, and if it is: Why was it stated in that question that MW is deterministic?
 A: In the many worlds interpretation, measurement devices (i.e. including things like the "conscious" (whatever that means) you) are equally part of the quantum system along with the putative "quantum system" that is under your measurement. 
During the "measurement", nothing remarkable happens: the whole quantum system (i.e. you and the studied system) remain in a pure (albeit monstrously high dimensional) quantum state. The state of this composite system evolves unitarily and utterly deterministically. The "you" and "measured" subsystem become entangled by the measurement, but this entanglement is still an outcome of deterministic pure state evolution.
So the point is that you and the measured system never come out of quantum superposition. All the base states (different "Worlds") are still present and in quantum superposition.
Now, as to which "World" "you" "see", that is quite another matter, and one which science has no accepted answer for at present. This is because science has no accepted precise description for consciousness and thus no description for your (nor mine, nor anyone else's) subjective experience; it has no consistent conception of "you", "I", nor any other personhood. 
So the simple answer to your question is that "science does not know". There are workers actively thinking about these ideas, but nothing has yet emerged as being remotely like "scientifically accepted".

User Innisfree comments / asks:

This is nothing to do with consciousness etc. The question is really, how to derive probabilities of observing various outcomes in QM (i.e Born rule) in MWI? 

This is an insightful reinterpretation of the OP's original question and one we can make some possible headway on. I am not an expert in MWI, but I understand that something approaching a possible explanation for the seeming emergence of a small number of possible states from a measurement even though the whole system evolves unitarily and deterministically is as follows. The interaction of the studied system with an external system will force the whole to undergo deterministic, unitary evolution determined by a Hamiltonian of the form $H_S + H_M + H_I$, where $H_S$ and $H_M$ are the Hamiltonians of the studied and measurement systems in isolation and $H_I$ is an interaction term. As outlined by user Daniel Sank in his answer here, some models of the interaction Hamiltonian $H_I$ reproduce the interesting behavior that, if the measurement system has a large number of unknown degrees of freedom, then then the result of the unitary evolution tends to be towards a pure state wherein the studied system part is near an eigenvector of a certain Hermitian operator, one of the so called tensor factors the interaction Hamiltonian $H_I$. Different eigenvectors are "selected" (so called einselection) by different initial states of the measurement system. The "probability" of selection of each eigenvector in this model (i.e. the proportion of unknown measurement system states, assuming each is equally likely) is given by the Born rule. But take special heed that there is no randomness here and no wavefunction collapse. This is a unitary state evolution brought about by the presence of the composite system.
A: You will see all outcomes, because "you" are part of the universe.
The idea that it is random is actually more in line with the Copenhagen interpretation than MWI.
The real issue is a philosophical one.  You are used to defining the concept of "you" in a way that makes sense in a classical world.  The "you" which makes sense in a MWI point of view must adapt to fit the model it is being used in.
After an event happens, you could choose to define a new observer in a random world, and declare that it embodies the perdurable concept of "you" from that point on.  However, the randomness was your own doing, not MWI.
From a physics perspective, there's just a bag of particles which are in different states in different worlds.  There's no metaphysical "you" that is observing the world from inside that bag.  How you adapt this metaphysical concept to fit within MWI is beyond the scope of physics.
A: Your question assumes there is something supernatural about consciousness. If you accept that our minds are just computers made up of biological components then there is no real mystery. Each copy of you will see the Universe they are in.
A: I think Rod Vance's answer is strictly correct. Science does not know how consciousness works and so it has no definitive stance on how our consciousness and our sense of self traverses the physical world (including the quantum world). That being said, I would argue that there are good working hypotheses about consciousness that are commonly held. In particular the hypothesis that consciousness is a product of the physical dynamics of the brain.
With this assumption in hand I think enough can be said to answer Deltab's question in full. In fact Deltab's question is not strictly confined to MWI. Rather it is about the unity of self in time. If one assumes that the self is unified in time, and then a physical theory predicts more than one copy of your body (as MWI does), then of course the natural question is how does nature spontaneously break the symmetry so that self remains unified in time. The simplest answer, in my opinion, is that if we assume that the sense of self is a product of the physical dynamics of the brain, then the assumption that self is unified in time is simply untenable. Each copy has equal right to the "self" because the physical dynamics of the split is symmetric.
I had considered using the example of the Star Trek teleporter, but perhaps instead we should consider a real experiment. The classic example of dis-unified self is in brain bisection operations. In these operations the corpus callosum of the brain is cut, leaving the two hemispheres of the brain unable to easily communicate, in control of different parts of the body, and receptive to different sense organs. Using these facts, researchers have been able to produce a significant amount of evidence suggesting that each hemisphere of the brain can operate independently of the other, including possessing independent knowledge and intent. This suggests that upon division of the brain, the self associated with the whole brain can become dis-unified into a pair of selves associated with each hemisphere of the brain. If I am about to undergo a brain bisection operation, the question of which hemisphere do "I" end up in after the surgery is fundamentally the same as the question of which world "I" will end up in after a quantum measurement, and the answer is that "you" will end up in each.
This poses difficult problems for probability theory, which I think are at the core of the challenge of understanding quantum mechanics. Namely, suppose instead of two brain hemispheres I had three tritospheres. Suppose I underwent a similar operation, only instead of cutting my brain in half, the operation cut my brain into two-thirds and one-third. Am I justified in assigning unequal probabilities now to which brain section I should expect to find myself in? Surely if what I've claimed prior is true, then that question is ill posed! Can we frame it in terms of decision theory? This I think remains a deep mystery.
