  We consider the direction set determined by various subsets $E$ of Euclidean
space and show that there is a trichotomy: Either (i) The subset is the graph
of a Lipschitz function and the direction set is not dense in the sphere, (ii)
The subset is the graph of a non-Lipschitz function and the direction set is
dense but not everything, or (iii) The subset is not a graph (in a suitable
sense) and every direction is determined by the set. We then explore a variety
of results based on this trichotomy under additional assumptions on the set
$E$.
