A singular foliation on a complex manifold M is determined by an involutive coherent subsheaf E of the tangent sheaf of M. In this paper, we study the problem of local (analytical and topological) triviality of the singular foliation along (a subset of) its singular set S(E). In general, S(E) is an analytic variety, so we examine by stratifying the set. For stratified subsets or stratified maps, the local topological triviality has been studied by a number of people and it is generally known that if the stratification satisfies the "Whitney condition" or the "Thom condition", then we have the local topological triviality along each stratum (the Isotopy Lemmas of Thom).
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