  In order to study projections of smooth curves, we introduce multifiltrations
obtained by combining flags of osculating spaces. We classify all
configurations of singularities occurring for a projection of a smooth curve
embedded by a complete linear system away from a projective linear space of
dimension at most two. In particular, we determine all configurations of
singularities of non-degenerate degree d rational curves in $\mathbb{P}^n$ when
$d - n \leq 3$ and $d < 2n$. Along the way, we describe the Schubert cycles
giving rise to these projections.
  We also reprove a special case of the Castelnuovo bound using these
multifiltrations: under the assumption $d < 2n$, the arithmetic genus of any
nondegenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d - n$.
