  We classify the canonical threefold singularities that allow an effective
two-torus action. This extends classification results of Mori on terminal
threefold singularities and of Ishida and Iwashita on toric canonical threefold
singularities. Our classification relies on lattice point emptiness of certain
polytopal complexes with rational vertices. Scaling the polytopes by the least
common multiple $k$ of the respective denominators, we investigate
$k$-emptiness of polytopes with integer vertices. We show that two dimensional
$k$-empty polytopes either are sporadic or come in series given by Farey
sequences. We finally present the Cox ring iteration tree of the classified
singularities, where all roots, i.e. all spectra of factorial Cox rings, are
generalized compound du Val singularities.
