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 the experimentally observed zero-bias anomalies are indeed due to 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 nontopological variants of the same system. A definitive and critical distinction between signals expected in topological (i.e., anyonic) and nontopological (i.e., trivial) situations is therefore essential for future progress in the field. In this paper, we carefully examine the expected signals of proposed braiding and fusion experiments in topological and nontopological variants of the experimental nanowire systems in which Majoranas are predicted to occur. We point out situations where trivial and anyonic signatures may be qualitatively 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 nontopological systems than fusion experiments or experiments with an odd number of Majorana exchanges. Successful fusion experiments, particularly in nanowires where zero bias conductance peaks are also observed, can also provide strong evidence for the existence of Majorana modes, but such fusion evidence without a corresponding braiding success is not definitive.