We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order $k$ foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter $k$ encodes the "order of tangency" of the leafwise vector fields to $L$. We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of $(k-1)$-jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.
Comment: Four paragraphs were added to the introduction which explain the relationship of the paper to work of Scott, Bischoff-del Pino-Witte, and Fischer-Laurent-Gengoux and describe the organization of the paper. The results are unchanged. 53 pages, 5 figures, 1 table