Every planar graph has at most $3n-6$ edges, where $n$ is the number of vertices. Moreover, every planar graph can be drawn with straight-line edges in the plane, without crossings. For example, for the complete graph $K_4$ one actually need 6 segments, so its segment number in 6. For any path its segment number is 1.
Dujmović, Eppstein, Suderman and Wood (2006) have shown that every planar graph can be drawn with at most $\frac52 n$ segments (where $n$ is the number of vertices).
Can this be improved?
Remark. There are examples of planar graphs that need $2n\pm O(1)$ segments.
The problem was asked on 24.03.2019 by Alex Ravsky and Alexander Wolff on pages 94-95 of Volume 2 of the Lviv Scottish Book.
Fig.1 in Dujmović et al. (a) shows a path has segment number $1$.