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In order to identify the sources of the observed diffuse high-energy neutrino flux, it is crucial to discover their electromagnetic counterparts. IceCube began releasing alerts for single high-energy ($E > 60$ TeV) neutrino detections with sky localisation regions of order 1 deg radius in 2016. We used Pan-STARRS1 to follow-up five of these alerts during 2016-2017 to search for any optical transients that may be related to the neutrinos. Typically 10-20 faint ($m < 22.5$ mag) extragalactic transients are found within the Pan-STARRS1 footprints and are generally consistent with being unrelated field supernovae (SNe) and AGN. We looked for unusual properties of the detected transients, such as temporal coincidence of explosion epoch with the IceCube timestamp. We found only one transient that had properties worthy of a specific follow-up. In the Pan-STARRS1 imaging for IceCube-160427A (probability to be of astrophysical origin of 50 %), we found a SN PS16cgx, located at 10.0' from the nominal IceCube direction. Spectroscopic observations of PS16cgx showed that it was an H-poor SN at z = 0.2895. The spectra and light curve resemble some high-energy Type Ic SNe, raising the possibility of a jet driven SN with an explosion epoch temporally coincident with the neutrino detection. However, distinguishing Type Ia and Type Ic SNe at this redshift is notoriously difficult. Based on all available data we conclude that the transient is more likely to be a Type Ia with relatively weak SiII absorption and a fairly normal rest-frame r-band light curve. If, as predicted, there is no high-energy neutrino emission from Type Ia SNe, then PS16cgx must be a random coincidence, and unrelated to the IceCube-160427A. We find no other plausible optical transient for any of the five IceCube events observed down to a 5 limiting magnitude of mag, between 1 day and 25 days after detection.