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We show that the monopole currents which one obtains in the maximally Abeliangauge of SU(2) fall into two quite distinct classes (when the volume is largeenough). In each field configuration there is precisely one cluster thatpermeates the whole lattice volume. It has a current density and a magneticscreening mass that scale and it produces the whole of the string tension. Theremaining clusters have a number density that follows an approximate power lawproportional to the inverse cube of l where l is the length of the monopoleworld line in lattice units. These clusters are localised in space-time withradii which vary as the square root of l. In terms of the radius r these`lumps' have a scale-invariant distribution proportional to (dr/r . 1/{r^4}).Moreover they appear not to contribute at all to the string tension. The factthat they are scale-invariant at small distances would seem to rule out aninstanton origin.Comment: LaTeX, 31 pages, 11 PostScript figures. Typo in Table 2 correcte

Monopole clusters in Abelian projected gauge theories