  We present various facts on the graded Betti table of a projectively embedded
toric surface, expressed in terms of the combinatorics of its defining lattice
polygon. These facts include explicit formulas for a number of entries, as well
as a lower bound on the length of the linear strand that we conjecture to be
sharp (and prove to be so in several special cases). We also present an
algorithm for determining the graded Betti table of a given toric surface by
explicitly computing its Koszul cohomology, and report on an implementation in
SageMath. This works well for ambient projective spaces of dimension up to
roughly $25$, depending on the concrete combinatorics, although the current
implementation runs in finite characteristic only. As a main application we
obtain the graded Betti table of the Veronese surface $\nu_6(\mathbb{P}^2)
\subseteq \mathbb{P}^{27}$ in characteristic $40\,009$. This allows us to
formulate precise conjectures predicting what certain entries look like in the
case of an arbitrary Veronese surface $\nu_d(\mathbb{P}^2)$.
