  In earlier work, we introduced the `Monster tower', a tower of fibrations
associated to planar curves.
  We constructed an algorithm for classifying its points with respect to the
equivalence relation generated by the action of the contact pseudogroup on the
tower.
  Here, we construct the analogous tower for curves in $n$-space. (This tower
is known as the Semple Bundle in Algebraic Geometry.) The pseudo-group of
diffeomorphisms of $n$-space acts on each level of the extended tower. We take
initial steps toward classifying points of this extended Monster tower under
this pseudogroup action.
  Arnol'd's list of stable simple curve singularities plays a central role in
these initial steps. We end with a list of open problems.
