  Given recent progress in the realization of Majorana zero modes in
semiconducting nanowires with proximity-induced superconductivity, a crucial
next step is to attempt an experimental demonstration of the predicted braiding
statistics associated with the Majorana mode. Such a demonstration should, in
principle, confirm that observed zero-bias anomalies are indeed indicative of
the presence of anyonic Majorana zero modes. Moreover, such a demonstration
would be a breakthrough at the level of fundamental physics: the first clear
demonstration of a non-Abelian excitation. It is therefore important to clarify
the expected signals of Majorana physics in the braiding context, and to
differentiate these signals from those that might also arise in non-topological
variants of the same system. A definitive and critical distinction between
signals expected in topological (i.e. anyonic) and non-topological (i.e.
trivial) situations is therefore essential for future progress in the field. In
this manuscript, we carefully examine the expected signals of braiding and
fusion experiments in topological and non-topological variants of the
experimental nanowire systems in which Majoranas are predicted to occur. We
point out situations where `trivial' and `anyonic' signatures may be similar
experimentally, necessitating a certain level of caution in the interpretation
of various proposed fusion and braiding experiments. We find in particular that
braiding experiments consisting of full braids (two Majorana exchanges) are
better at distinguishing between topological and non-topological systems than
fusion experiments or experiments with an odd number of Majorana exchanges.
