The cofinality of a totally ordered set is always a regular cardinal. On the other hand for any cardinal (regular or singular) $\kappa$ there is a partially ordered set $(A,\leq)$ with $\operatorname{cf}(A,\leq)=\kappa$, for example $(\kappa,=)$.
Assume that $(A,\leq)$ is a partially ordered set of infinite cofinality $\kappa$, and furthermore that $(A,<)$ is $\kappa$-directed, meaning that if $B\subset A$ and $|B|<\kappa$ then there is $a\in A$ such that $b for all $b\in B$. With this additional assumption, is it still possible that $\kappa$ is a singular cardinal?
Added question: If we change the additional assumption of $\kappa$-directedness to the assumption that $(A,\leq)$ is a directed set, is it even then possible that $\operatorname{cf}(A,\leq)$ is a singular cardinal? In other words, is there a directed partially ordered set whose cofinality is a singular cardinal?