Mark_Friedenbach comments on Probability is in the Mind - Less Wrong
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It's not just about collapse - every single-world QM interpretation either involves extra postulates, non-locality or other surprising alterations of physical law, or yields falsified predictions. The FAQ I linked to addresses these points in great detail.
MWI is simple in the Occam's razor sense - it is what falls out of the equations of QM if you take them to represent reality at face value. Single-world meta-theories require adding additional restrictions which are at this time completely unjustified from the data.