  A number of recent works have discussed the issue of spin polarization of a
Majorana zero mode in condensed matter systems. Here we show that the spin
polarization density of a Majorana zero mode, computed as an average of the
spin operator over its wave function, vanishes everywhere. A single
non-degenerate Majorana zero mode, therefore, does not couple to an applied
magnetic field, except via hybridization with higher energy excited states (if
present), which may perturb its wave function. If `spin' is defined by
considering only the particle components of the wave function, as has been done
in some recent works, Majorana zero modes do have a non-zero spatial profile of
this quantity, measurable in scanning tunneling microscopy (STM) experiments.
However, if such a quantity is measured in spin-resolved tunneling experiments
(without spatial resolution), we show that it cannot be used as a unique
signature of Majorana zero modes in the topologically non-trivial
superconducting phase. As a byproduct, we show that in spatially inhomogeneous
systems, accidental zero energy modes, which for all practical purposes behave
as Majorana zero modes (including giving rise to a zero bias conductance peak
of height 2e^2/h), can appear with increasing magnetic field even in the
absence of a topological quantum phase transition (TQPT). But only after gap
closing and the associated TQPT, the modes are localized near the system edges,
resulting in the maximum topological protection. In the light of these
considerations, demonstrating the nonlocal character of the
topologically-protected Majorana pair and its emergence {\em after} the systems
undergo a TQPT, become critical tasks for the ongoing experimental search for
Majorana bound states in condensed matter systems.
