  The purpose of this article is to give new constructions of linear orders
which are minimal with respect to being non-$\sigma$-scattered. Specifically,
we will show that Jensen's principle $\diamondsuit$ implies that there is a
minimal Countryman line, answering a question of Baumgartner. We also produce
the first consistent examples of minimal non-$\sigma$-scattered linear orders
of cardinality greater than $\aleph_1$, as given a successor cardinal
$\kappa^+$, we obtain such linear orderings of cardinality $\kappa^+$ with the
additional property that their square is the union of $\kappa$-many chains. We
give two constructions: directly building such examples using forcing, and also
deriving their existence from combinatorial principles. The latter approach
shows that such minimal non-$\sigma$-scattered linear orders of cardinality
$\kappa^+$ exist for every cardinal $\kappa$ in G\"odel's constructible
universe, and also (using work of Rinot) that examples must exist at successors
of singular strong limit cardinals in the absence of inner models satisfying
the existence of a measurable cardinal $\mu$ of Mitchell order $\mu^{++}$.
