Stefan-Sussmann singular foliations, singular subalgebroids and their associated sheaves

We explain and motivate Stefan-Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan-Sussmann singular foliation or singular subalgebroid.