A linear order is scattered if it does not contain a copy of the rational line and $\sigma$-scattered if it can be decomposed into countably many scattered linear orders. Laver has shown that the class of $\sigma$-scattered linear orders is well quasi-ordered and hence well understood. It is therefore natural to study linear orders which are minimal with respect to being non-$\sigma$-scattered---those which embed into all of their non-$\sigma$-scattered suborders. We give the first consistent construction of a minimal non-$\sigma$-scattered linear orders of cardinality greater than $\aleph_1$ and also show that $\diamondsuit$ implies that there is a minimal non-$\sigma$-scattered linear order which is a Countryman line. This is joint work with Todd Eisworth and James Cummings.