  Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to
show that any tangential homotopy can be approximated by a homotopy of
topological embeddings with mild singularities. This concept plays an important
role in Contact Topology: The loose legendrian h-principle of E. Murphy relies
on wrinkled embeddings to manipulate the legendrian front. Similarly, the
simplification of legendrian front singularities was proven by D.
\'Alvarez-Gavela by defining the notion of wrinkled legendrian.
  This paper and its sequel generalise these ideas to general jet spaces. The
main theorem in the present paper proves the analogue of the result by
Eliashberg and Mishachev: Any homotopy of the r-order differential information
of an embedding can be approximated by a homotopy of embeddings with
wrinkle-type singularities (of order r).
  The local version of the previous statement, which is of independent
interest, says that the holonomic approximation theorem holds over closed
manifolds if, instead of sections, we consider multi-valued sections with
simple singularities.
