  We examine neutrino zeromode solutions on the electroweak $Z$-string and
their effect on the stability of the string in the standard model and its
extensions. We propose using topological invariants constructed from the
momentum (and real) space topology of Green's functions, often used for
investigating edge modes in condensed matter physics. We analyze the standard
model and then examine type-I and type-II extensions of the neutrino sector as
well as their hybrid. Based on this analysis, we also comment on proposals in
the literature to stabilize the $Z$-string.
