So by a theorem of Solovay we know that a stationary subset of a regular (uncountable) cardinal $\kappa$ can always be split into $\kappa$-many stationary subsets.
Is the regularity assumption indispensable?
What if $\kappa$ is singular? Does the conclusion necessarily fail or are there known examples of singular $\kappa$ for which it still holds?
The regularity is necessary, but you can circumvent it as follows:
First of all, for singular cardinals of countable cofinality the usual notion for clubs is meaningless, since there are $\omega$-sequences which are unbounded, so you can easily avoid anything which does not contain a tail-segment of your cardinal.
Secondly, for uncountable cofinality singular cardinals, there is a club of order type $\operatorname{cf}(\kappa)$. So every stationary set can be reduced to be meaningful only on that club, which is small. And you cannot partition a small set into too many disjoint sets. But you can pull that stationary set into a stationary subset of $\operatorname{cf}(\kappa)$, which is regular (and uncountable), split it there, and push that back to stationary subsets of $\kappa$.