  We prove that a generic canonically or bicanonically embedded smooth curve
has semistable m-th Hilbert points for all m. We also prove that a generic
bicanonically embedded smooth curve has stable m-th Hilbert points for all m
\geq 3. In the canonical case, this is accomplished by proving finite Hilbert
semistability of special singular curves with G_m-action, namely the
canonically embedded balanced ribbon and the canonically embedded balanced
double A_{2k+1}-curve. In the bicanonical case, we prove finite Hilbert
stability of special hyperelliptic curves, namely Wiman curves. Finally, we
give examples of canonically embedded smooth curves whose m-th Hilbert points
are non-semistable for low values of m, but become semistable past a definite
threshold.
  (This paper subsumes the previous submission and arXiv:1110.5960).
