Between 2011 March and 2014 August Swift responded to 20 triggers from the IceCube neutrino observatory, observing the IceCube 50 per cent confidence error circle in X-rays, typically within 5 h of the trigger. No confirmed counterpart has been detected. We describe the Swift follow-up strategy and data analysis and present the results of the campaign. We discuss the challenges of distinguishing the X-ray counterpart to a neutrino trigger from serendipitous uncatalogued X-ray sources in the error circle, and consider the implications of our results for future strategies for multimessenger astronomy, with particular reference to the follow-up of gravitational wave triggers from the advanced-era detectors.