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Background Stepped-wedge cluster randomised trials (SW-CRTs) are a pragmatic trial design, providing an unprecedented opportunity to increase the robustness of evidence underpinning implementation and quality improvement interventions. Given the complexity of the SW-CRT, the likelihood of trials not delivering on their objectives will be mitigated if a feasibility study precedes the definitive trial. It is not currently known if feasibility studies are being conducted for SW-CRTs nor what the objectives of these studies are. Methods Searches were conducted of several databases to identify published feasibility studies which were designed to inform a future SW-CRT. For each eligible study, data were extracted on the characteristics of and rationale for the feasibility study; the process for determining progression to the main trial; how the feasibility study informed the main trial; and whether the main trial went ahead. A narrative synthesis and descriptive analysis are presented. Results Eleven feasibility studies were identified, which included eight completed study reports and three protocols. Three studies used a stepped-wedge design and these were the only studies to be randomised. Studies were predominantly of a mixed-methods design. Only one study assessed specific features related to the feasibility of using a SW-CRT and one investigated the time taken to complete the study procedures. The other studies were mostly assessing the feasibility and acceptability of the intervention. Conclusion Published feasibility studies for SW-CRTs are scarce and those that are being reported do not investigate issues specific to the complexities of the trial design. When conducting feasibility studies in advance of a definitive SW-CRT, researchers should consider assessing the feasibility of study procedures, particularly those specific to the SW-CRT design, and ensure that the findings are published for the benefit of other researchers. Electronic supplementary material The online version of this article (10.1186/s12874-019-0658-3) contains supplementary material, which is available to authorized users.