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We determine the thermodynamic stability conditions for near-extreme rotatingD3, M5, and M2-branes with multiple angular momenta. Critical exponents nearthe boundary of stability are discussed and compared with a naive field theorymodel. From a partially numerical computation we conclude that outside theboundary of stability, the angular momentum density tends to become spatiallyinhomogeneous.  Periodic Euclidean spinning brane solutions have been studied as models ofQCD. We explain how supersymmetry is restored in the world-volume field theoryin the limit of large spin and discuss the hierarchy of energy scales thatdevelops as this limit is approached.Comment: 37 pages, 4 figures. v2: revised and reordered to improve clarity,  version to appear in JHE

Thermodynamic stability and phases of general spinning branes