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We present an in-depth analysis of the flavour and spectral composition ofthe 36 high-energy neutrino events observed after three years of observation bythe IceCube neutrino telescope. While known astrophysical sources of HEneutrinos 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 theevents detected above $E_\nu \ge 28$ TeV do not necessarily support thishypothesis. Crucially, the energy range that is considered when analysing theHE neutrino data can have a profound impact on the conclusions. We highlighttwo intriguing puzzles: an apparent deficit of muon neutrinos, seen via adeficit of track-like events; and an absence of $\bar \nu_e$'s at high energy,seen as an absence of events near the Glashow resonance. We discuss possibleexplanations, including the misidentification of tracks as showers, and abroken power law, in analogy to the observed HE cosmic ray spectrum.

The Flavour Composition of the High-Energy IceCube Neutrinos