Many-worlds: how often is the split how many are the universes? (And how do you model this mathematically.) When I read descriptions of the many-worlds interpretation of quantum mechanics, they say things like "every possible outcome of every event defines or exists in its own history or world", but is this really accurate? This seems to imply that the universe only split at particular moments when "events" happen. This also seems to imply that the universe only splits into a finite number of "every possible outcome". 
I imagine things differently. Rather than splitting into a finite number of universes at discrete times, I imagine that at every moment the universe splits into an uncountably infinite number of universes, perhaps as described by the Schrödinger equation.
Which interpretation is right? (Or otherwise, what is the right interpretation?) If I'm right, how does one describe such a vast space mathematically? Is this a Hilbert space? If so, is it a particular subset of Hilbert space?
 A: The Many Worlds interpretation is popularly misunderstood. The wave function itself contains a spectrum of universes, one corresponding to each eigenvalue for a given operator. The "splitting" of the "many worlds" is represented by the time evolution of the wave function described by the Schrodinger equation. As Lubos mentions above, these "universes" only become separate through decoherence.
Consider, for example, a wave function in the position-basis given by a delta-function at x=0. This represents one universe. Now time-evolve the wave function using the schrodinger equation. The delta-function has now spread-out a bit. It is peaked at x=0, but has non-zero values at x=+1 and x=-1. This represents the existence of universes in which the position of the particle is at x=0, x=+1, and x=-1. In some sense there are "more" universes at x=0 than at x=+-1, because the wave function is more highly peaked at x=0. This is where some of the difficulty in the Many Worlds interpretation comes in: what ontology to use to describe the "splitting", "how many universes" are at x=0 vs x=+-1, and so on. The main point I want to make is that the "splitting" is just an interpretation of what is happening with the evolution of the wave function according to the schrodinger equation. Nothing "more" is actually happening. You model the "splitting" using the tried-and-true schrodinger evolution of the wave function.
A: Many worlders won't tell you this dirty little secret but how often splitting happens, and how many worlds there are, depends upon the choice of coarse graining, and the coarse graining resolution. No, it's not possible to ramp up the coarse graining all the way to the finest levels because a decoherence/coherence threshold would be crossed. And no, there is no canonical coarse graining either.
The preferred basis depends upon the environment. Always. What is the preferred basis for a closed self-contained universe?
A: An interesting many-worlds model is the "many interacting worlds" (MIW) in which there is no "splitting" (creation or destruction of worlds) at all. Rather, worlds influence each other when they are near each other in configuration space (their constituent particles are all nearly at the same position and orientation). Each world is strictly Newtonian, except for this interaction which can (amazingly to me) reconstruct the well-known Schordinger quantum phenomenon, including the uncertainty principle and interference patterns, etc. See http://dx.doi.org/10.1103/PhysRevX.4.041013
A conceptual burden one must bear in holding onto this MIW model is the unknown inter-world interaction force, which must be repulsive and satisfy certain constraints but is otherwise open to be freely chosen. 
On the other hand, there is no need to deal with "collapse," so a matter of taste if this burden is a net win.
A: The answer is 'it depends'.  It depends on what you mean by "world".  Personally, I prefer the word "timeline" , but it makes no difference.  And before I give my answer I should state that Hugh Everett, the formulator of MWI, reckoned there were an uncountable infinity of worlds (or timelines).  Unfortunately he did not state his reasoning for this, since it was just a remark given in passing, in an answer to a conference question, c.1962.
Everett did state in his thesis that the branching of an observer was limited by the observer's entropy or memory capacity.  Since these are finite quantities we would expect the branching to be finite, although usually very large.  (Unless of course the universe is infinitely large, in which case its entropy is as well, but I don't think Everett was referring to this.)
There is also Boltzmann's S=k log W to consider, which again suggest the number of worlds (now each considered as a microstate) is finite.  Although the infra red divergences suggest it may be infinite, since the number of real photons emitted during Bremsstrahlung is infinite - although only a finite number can be measured..
So, on balance, I would say the branching is finite, if we think in terms of the number of possible distinct configurations of any system, including any observer such as you and me.  But the number of branches of the totality of existence could well be infinite.
How do we model splitting mathematically? After a measurement the observer can be deemed to be split if the overlap (which has a precise technical definition) between the different observer states is negligible.
