We present an in-depth analysis of the flavour and spectral composition of the 36 high-energy neutrino events observed after three years of observation by the IceCube neutrino telescope. While known astrophysical sources of HE neutrinos are expected to produce a nearly $(1:1:1)$ flavour ratio (electron : muon : tau) of neutrinos at earth, we show that the best fits based on the events detected above $E_ u ge 28$ TeV do not necessarily support this hypothesis. Crucially, the energy range that is considered when analysing the HE neutrino data can have a profound impact on the conclusions. We highlight two intriguing puzzles: an apparent deficit of muon neutrinos, seen via a deficit of track-like events; and an absence of $bar u_e$s at high energy, seen as an absence of events near the Glashow resonance. We discuss possible explanations, including the misidentification of tracks as showers, and a broken power law, in analogy to the observed HE cosmic ray spectrum.
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