  We study the formation of lines in phase space in Wigner's distribution $W.$
Whereas lines in phase space do not form in classical systems, unless special
initial states are chosen, we find, for large classes of systems and initial
states of quantum systems that $W$ tends to form straight line patterns
crisscrossing phase space. These arise from the states' coherences. Some of
those lines have astonishing extent, reaching across the entire state. We show
that the formation of such straight line patterns is due to the formation of
'randomized grid states'. We establish their stability to perturbations, and
that they are tied to interference phenomena in configuration space. We
additionally identify generic higher-order `eye' patterns in phase space which
occur less often since they require the formation of more specific regular grid
states; and we show that the randomization of eye patterns tends to deform them
into lines.
