The order-theoretic structure of the ordinal numbers in non-standard models I have read the following.

Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a copy of $\mathbb{N}$ followed by $\mathbb{Q}$-many copies of $\mathbb{Z}$.

Now assume there exists an inaccessible cardinal, call it $\kappa$, and let $T$ denote the first-order theory of $(V_\kappa,\in)$. Let $C$ denote a non-well-founded $T$-model that is countable, and $U$ denote a non-well-founded $T$-model having cardinality $\kappa,$ so that in particular it is the same cardinality as $V_\kappa.$
Can we make any statements about the order-theoretic structure of $\mathrm{On}^C$ and/or $\mathrm{On}^U$?
 A: The order types of ill-founded models were described, to an extent, by Friedman.
He mimics the analysis of PA models using Cantor normal forms. There are essentially two cases, depending on whether or not $M$ is an $\omega$-model. If it is then the Cantor normal forms are relatively well behaved; you can partition the ordinals of $M$, as in the PA case, into blocks of length the well-founded height of $M$ and these blocks form a dense linear order. So in the case that $M$ is an $\omega$-model of well-founded height $\alpha$, the ordinals of $M$ have order type $\alpha\cdot(1+\eta)$ where $\eta$ is some dense linear order. In particular, if $M$ is countable this is just $\alpha\cdot (1+\mathbb{Q})$.
If $M$ isn't an $\omega$-model the blocks we used above might not all have the same length and things get more complicated. Nevertheless, Friedman gives something of a description of the order type here as well. In particular, if $M$ is countable then the order type is determined by the standard system of $M$, i.e. the collection of intersections of elements of $M$ with the (collapse of the) well-founded part of $M$.
Saying anything more that this for structures of higher cardinality would require saying more about dense linear orders of that cardinality. Hutchinson has managed to obtain nice descriptions for $\aleph_1$-like models, but I doubt much more is known since the question of order types in higher cardinalities is a major open problem even for models of PA, let alone set theory.

H. Friedman, Countable models of set theories, Cambridge Summer School in Mathematical Logic, 1971, Lecture Notes in Mathematics, Springer, 337 (1973), 539--573
J.E. Hutchinson, Order types of ordinals in models of set theory, J. Sym. Logic 41(2), 1976, 489--502

