Are mantle plumes distributed around the core randomly or in a known pattern?
First of all, the idea of a fixed "hotspot" reference frame is (albeit reluctantly) falling out of favour on a geological timescale; see e.g. http://onlinelibrary.wiley.com/doi/10.1029/GM121p0339/summary ("As studies of plate motions have advanced, however, it has become clear that the global hotspots do not stay fixed relative to each other...") and http://adsabs.harvard.edu/abs/2010AGUFM.U51A0010S ("Hotspot reference frames can only be confidently tied back to about 130 Ma and there is evidence that mantle plumes have moved relative to each other.") Global Moving Hotspot Reference Frames are in the process of being built, see e.g. http://adsabs.harvard.edu/abs/2010AGUFMGP24A..03D

Secondly, identifiable patterns in the distribution of mantle plumes are an area of active research: this year's EGU medal for outstanding young scientist was awarded to Rhodri Davies for his work in this area. Key to most arguments are the presence of Large Low Shear Velocity Provinces (LLSVPs) in the lower mantle, historically assumed to be "slab graveyards" (thermochemical piles created when subducting slabs pool in the lower mantle). The suggestion that these areas were both chemically and thermally distinct from surrounding mantle focussed attention on their margins as a highly probable location for the formation of mantle plumes. However, recent findings (by and cited extensively by Rhodri Davies) increasingly support the idea that these piles are thermal features only, with no chemical difference to surrounding mantle.

Rhodri Davies' research revisits past studies that found hotspots were concentrated along the edges of LLSVPs, and has thoroughly demonstrated that these conclusions are not robust, e.g. to choice of hotspot catalogue, prescribed angular search tolerance (between surface expression and imaged margin), (arbitrary) depth of analysis, and (arbitrary) shear-wave velocity contour. Via Monte Carlo simulations he rules out hotspots being drawn from the margins of the Pacific LLSVP, but cannot rule out the possibility that hotspots are preferentially drawn from the entire areal extent of LLSVPs in both the Pacific and the Atlantic.

RD &amp; his team conclude that present-day hotspot locations are controlled by ancient subduction, rather than the current margins of LLSVPs - which are demonstrated via dynamical simulations to be mobile, even in the presence of dense chemical heterogeneity.

(As a side note, both "hotspots" and "mantle plume" are contested terminology, it's true - but the vast majority of geoscientists believe that mantle plumes do exist. There is complexity to that view, in that the consensus does not require all intraplate volcanism to be the result of atypically hot mantle -- but the idea that mantle plumes simply do not exist, as espoused by mantleplumes.org, does not have wide acceptance.)