Holomorphic foliations with singularities are considered. In this context, a singularity, or a singular locus, can be described as follows: in a fiber bundle \(\{\) \(F\to E\to B\}\) (with appropriate structure), or in a sheaf over B (with appropriate structure), let a section X be given over \(B-S_ 0\), with \(\overline{B-S_ 0}=B\). Then, in general, the closure \(\bar X\) in the total space E is a section with singularity \(S=\bar X-X\) over \(S_ 0\). A foliation of a manifold M defines a section in a Grassmann bundle over M, and this leads to a treatment of foliations with singularities. The Nash and Grassmann graph constructions, e.g. the blow-up process, are studied by the author. A geometric view of the Baum- Bott residues is presented. The rationality conjecture of \textit{P. Baum} and \textit{R. Bott} [J. Diff. Geom. 7, 279-342 (1972; Zbl 0268.57011)] is reduced to the domain of vector bundles in case the singular holomorphic foliation is given by the image sheaf of a bundle morphism. If the rationality conjecture holds for vector bundles that produce singular holomorphic foliations then it holds for integrable image sheaves of bundle morphisms. This is applied to the blow-up process.