The IceCube neutrino observatory pursues a follow-up program selecting interesting neutrino events in real-time and issuing alerts for electromagnetic follow-up observations. In 2012 March, the most significant neutrino alert during the first three years of operation was issued by IceCube. In the follow-up observations performed by the Palomar Transient Factory (PTF), a Type IIn supernova (SN IIn) PTF12csy was found 0.°2 away from the neutrino alert direction, with an error radius of 0.°54. It has a redshift of z = 0.0684, corresponding to a luminosity distance of about 300 Mpc and the Pan-STARRS1 survey shows that its explosion time was at least 158 days (in host galaxy rest frame) before the neutrino alert, so that a causal connection is unlikely. The a posteriori significance of the chance detection of both the neutrinos and the SN at any epoch is 2.2 σ within IceCube's 2011/12 data acquisition season. Also, a complementary neutrino analysis reveals no long-term signal over the course of one year. Therefore, we consider the SN detection coincidental and the neutrinos uncorrelated to the SN. However, the SN is unusual and interesting by itself: it is luminous and energetic, bearing strong resemblance to the SN IIn 2010jl, and shows signs of interaction of the SN ejecta with a dense circumstellar medium. High-energy neutrino emission is expected in models of diffusive shock acceleration, but at a low, non-detectable level for this specific SN. In this paper, we describe the SN PTF12csy and present both the neutrino and electromagnetic data, as well as their analysis.