  We prove asymptotic 0-1 Laws satisfied by diagrams of unimodal sequences of
positive integers. These diagrams consist of columns of squares in the plane,
and the upper boundary is called the shape. For various types, we show that, as
the number of squares tends to infinity, $100\%$ of shapes are near a certain
curve---that is, there is a single {\it limit shape}. Similar phenomena have
been well-studied for integer partitions, so the present work is a natural
extension. One notable corollary is a transferred limit shape for
overpartitions.
