  Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$
foliation on $M$, with isolated singularities of Morse type. The study and
classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In
this setting, a classical result due to Reeb \cite{Reeb} states that a manifold
admitting a foliation with exactly two center-type singularities is a sphere.
In particular this is true if the foliation is given by a function. Along these
lines a result due to Eells and Kuiper \cite{Ku-Ee} classify manifolds having a
real-valued function admitting exactly three non-degenerate singular points. In
the present paper, we prove a generalization of the above mentioned results. To
do this, we first describe the possible arrangements of pairs of singularities
and the corresponding codimension one invariant sets, and then we give an
elimination procedure for suitable center-saddle and some saddle-saddle
configurations (of consecutive indices). In the second part, we investigate if
other classical results, such as Haefliger and Novikov (Compact Leaf) theorems,
proved for regular foliations, still hold true in presence of singularities. At
this purpose, in the singular set, $Sing(\F)$ of the foliation $\F$, we
consider {\em{weakly stable}} components, that we define as those components
admitting a neighborhood where all leaves are compact. If $Sing(\F)$ admits
only weakly stable components, given by smoothly embedded curves diffeomorphic
to $S^1$, we are able to extend Haefliger's theorem. Finally, the existence of
a closed curve, transverse to the foliation, leads us to state a Novikov-type
result.
