  Sterile neutrinos in the mass range of a few keV are candidates for both cold
and warm dark matter. An ad-mixture of a heavy neutrino mass eigenstate to the
electron neutrino would result in a minuscule distortion - a 'kink' - in a
$\beta$-decay spectrum. In this paper we show that a wavelet transform is a
very powerful shape analysis method to detect this signature. For a tritium
source strength, similar to what is expected from the KATRIN experiment, a
statistical sensitivity to active-to-sterile neutrino mixing down to $\sin^2
\theta= 10^{-6}$ ($90\%$ CL) can be obtained after 3 years of measurement time.
It is demonstrated that the wavelet approach is largely insensitive to
systematic effects that result in smooth spectral modifications. To make full
use of this analysis technique a high resolution measurement (FWHM of
$\sim100$~eV) of the tritium $\beta$-decay spectrum is required.
