The ever-growing size of pretrained language models (PLM) presents a significant challenge for efficiently fine-tuning and deploying these models for diverse sets of tasks within memory-constrained environments.In light of this, recent research has illuminated the possibility of selectively updating only a small subset of a model’s parameters during the fine-tuning process.Since no new parameters or modules are added, these methods retain the inference speed of the original model and come at no additional computational cost. However, an open question pertains to which subset of parameters should best be tuned to maximize task performance and generalizability. To investigate, this paper presents comprehensive experiments covering a large spectrum of subset selection strategies. We comparatively evaluate their impact on model performance as well as the resulting model’s capability to generalize to different tasks.Surprisingly, we find that the gains achieved in performance by elaborate selection strategies are, at best, marginal when compared to the outcomes obtained by tuning a random selection of parameter subsets. Our experiments also indicate that selection-based tuning impairs generalizability to new tasks.