Summary: In [Publ. Math., Inst. Hautes Étud. Sci. 43, 47-100 (1974; Zbl 0279.58009)], F. Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions \(\geq 3\) he had to use the notion of ‘weak-\(C^0\)-equivalence’.
In this paper we show how to classify singularities of vector fields on \(\mathbb{R}^3\) up to codimension 4 for the notion of \(C^0 \) equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak-\(C^0\)-equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are \(C^\infty\), although it will be clear that the results are valid for \(C^r\), with \(r\) sufficiently big. We provide a complete, but short survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.