  The first detection of high-energy astrophysical neutrinos by IceCube
provides new opportunities for tests of neutrino properties. The long baseline
through the Cosmic Neutrino Background (C$\nu$B) is particularly useful for
directly testing secret neutrino interactions ($\nu$SI) that would cause
neutrino-neutrino elastic scattering at a larger rate than the usual weak
interactions. We show that IceCube can provide competitive sensitivity to
$\nu$SI compared to other astrophysical and cosmological probes, which are
complementary to laboratory tests. We study the spectral distortions caused by
$\nu$SI with a large s-channel contribution, which can lead to a dip, bump, or
cutoff on an initially smooth spectrum. Consequently, $\nu$SI may be an exotic
solution for features seen in the IceCube energy spectrum. More conservatively,
IceCube neutrino data could be used to set model-independent limits on $\nu$SI.
Our phenomenological estimates provide guidance for more detailed calculations,
comparisons to data, and model building.
