I've always seen Aronszajn trees being discussed on regular cardinals :
Let $\kappa$ be a regular infinite cardinal. A $\kappa$-Aronszajn tree is a tree on $\kappa$ of height $\kappa$, whose levels have all cardinality less than $\kappa$, and with no $\kappa$-branch.
Does this definition holds if $\kappa$ is singular? Do we have any reason for looking almost only at regular cardinals?
The definition is meaningful for $\kappa$ singular, but is not very interesting because $\kappa$-Aronszajn trees can easily be shown to exist in this case. If $\operatorname{cf}(\kappa) < \kappa$, take ordinals $\alpha_\xi < \kappa$ for $\xi < \operatorname{cf}(\kappa)$, such that $\sup_\xi \alpha_\xi = \kappa$, and consider the disjoint union of paths of length $\alpha_\xi$.