Let $\mathcal{T}\_\kappa$ be the set of all linear order types of cardinality $\kappa$. Let $\prec$ denote a binary relation on $\mathcal{T}\_\kappa$ representing embeddability of order types (note that $\prec$ is a quasi-order).


* What is the cardinality of a subset of $\mathcal{T}\_\kappa$ of the largest size that can be equipped with a linear order $<$ consistent with $\prec$ (such that $\tau<\mu$ implies $\tau\prec\mu$)?
* What is the cardinality of a subset of $\mathcal{T}\_\kappa$ of the largest size such that all its elements are mutually embeddable?


I am particularly interested in cases where $\kappa$ is $\aleph\_0$, $\aleph\_1$ and $\mathfrak{c}$.



Maximal chains in a quasi-order of linear order types
If you only consider countable linear orders, then for example the order types $\eta+2+\eta+2+\eta+3+\eta+\cdots$ and 
 $\eta+3+\eta+2+\eta+3+\eta+\cdots$ are not isomorphic. (Where $\eta$ is the order type of the rationals.)


This way you can get continuum many nonisomorphic orders, any two of which are bi-embedable. You can linearly order them in any way you want. 


ADDED: As Joel points out in his comment, this can be generalized to cardinals $\kappa$ satisfying $\kappa^{<\kappa}=\kappa$ as follows: 


Use the assumption $\kappa^{<\kappa}=\kappa$ to find a universal linear order of size $\kappa$ (i.e., one into which any other such order embeds). Without loss of generality you may assume that this order is dense; let us call this order $H$ (capital eta). Now consider well-ordered sums of length $\kappa$ in which $H$ alternates with 2-element and/or 3-element linear orders. All these orders will be non-isomorphic, because the places where the 2's and 3's appear can be recovered from the abstract order type, using the fact that $H$ is dense.


There are $2^\kappa$ many of these order types, and they are pairwise bi-embeddable. 


(I previously wrote $2^{<\kappa}=\kappa$ instead of $\kappa^{<\kappa}=\kappa$, but it is not obvious that this assumption suffices.)


