  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 $E_\nu \ge 28$ 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 $\bar \nu_e$'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.
