  We study the vanishing neighbourhood of non-isolated singularities of
functions on singular spaces by associating a general linear function.
  We use the carrousel monodromy in order to show how to get a better control
over the attaching of thimbles. For one dimensional singularities, we prove
obstructions to integer (co)homology groups and to the eigenspaces of the
monodromy via monodromies of nearby sections. Our standpoint allows to find, in
certain cases, the structure of the Milnor fibre up to the homotopy type.
