(This used to be my research area, but I am not longer active in this topic, so I don't know all the latest references). Nevertheless, last year X. Buff and A. Chéritat (see the X. Buff's preprint page) proved that there exist Julia sets with positive Lebesgue measure (a result which was presented at this years' ICM), which would lend credence to the conjecture that so does the Mandelbrot set. But that, AFAIK, is still open. Xavier and Arnaud would be the best people from whom to ask this question.
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