We answer a question of Brandstädt et al. by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is NP-complete. Conversely, the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently. The proof of our NP-completeness result is based on a refinement on a result due to Chvátal that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem. Here, a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We also discuss some open problems on stable cutsets.