  We present existence results for certain singular 2-dimensional foliations on
4-manifolds. The singularities can be chosen to be simple, e.g. the same as
those that appear in Lefschetz pencils. There seems to be a wealth of such
creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to
be transverse or be leaves of these foliations.
  The purpose of this paper is to offer objects, hoping for a future theory to
be developed on them. For example, foliations that are taut might offer genus
bounds for embedded surfaces (Kronheimer's conjecture).
