  We study notions of generic and coarse computability in the context of
computable structure theory. Our notions are stratified by the $\Sigma_\beta$
hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level
all linear orderings have both generically and coarsely computable copies. This
behavior changes abruptly at higher levels; we show that at the
$\Sigma_{\alpha+2}$ level for any $\alpha\in\omega_1^{ck}$ the set of linear
orderings with generically or coarsely computable copies is
$\mathbf{\Sigma}_1^1$-complete and therefore maximally complicated. This
development is new even in the general analysis of generic and coarse
computability of countable structures. In the process of proving these results
we introduce new tools for understanding generically and coarsely computable
structures. We are able to give a purely structural statement that is
equivalent to having a generically computable copy and show that every
relational structure with only finitely many relations has coarsely and
generically computable copies at the lowest level of the hierarchy.
