Cognitive stability, the ability to focus on a current task, and cognitive flexibility, the ability to switch between different tasks, are traditionally conceptualized as opposing end-points on a one-dimensional continuum. This assumption obligates a stability-flexibility trade-off - greater stability equates to less flexibility, and vice versa. In contrast, a recent cued task-switching study suggested that stability and flexibility can be regulated independently, evoking a two-dimensional perspective where trade-offs are optional (Geddert & Egner, Journal of Experimental Psychology: General, 151, 3009-3027, 2022). This raises the question of under what circumstances trade-offs occur. We here tested the hypothesis that trade-offs are guided by cost-of-control considerations whereby stability and flexibility trade off in contexts that selectively promote stability or flexibility, but not when neither or both are promoted. This proposal was probed by analyzing whether a trial-level metric of a stability-flexibility trade-off, an interaction between task-rule congruency and task sequence, varied as a function of a broader block-level context that independently varied demands on stability or flexibility by manipulating the proportion of incongruent and switch trials, respectively. In Experiment 1, we reanalyzed data from Geddert and Egner, Journal of Experimental Psychology: General, 151, 3009-3027, (2022); Experiment 2 was a conceptual replication with a design tweak that controlled for potential confounds due to local trial history effects. The experiments produced robust evidence for independent stability and flexibility adaptation, and for a context-dependent expression of trial-level stability-flexibility trade-offs that generally conformed to the cost-of-control predictions. The current study thus documents that stability-flexibility trade-offs are not obligatory but arise in contexts where either stability or flexibility are selectively encouraged.