  We investigate the effects of dissipation and driving on topological order in
superconducting nanowires. Rather than studying the non-equilibrium steady
state, we propose a method to classify and detect dynamical signatures of
topological order in open quantum systems. Bulk winding numbers for the
Lindblad generator $\hat{\mathcal{L}}$ of the dissipative Kitaev chain are
found to be linked to the presence of Majorana edge master modes -- localized
eigenmodes of $\hat{\mathcal{L}}$. Despite decaying in time, these modes
provide dynamical fingerprints of the topological phases of the closed system,
which are now separated by intermediate regions where winding numbers are
ill-defined and the bulk-boundary correspondence breaks down. Combining these
techniques with the Floquet formalism reveals higher winding numbers and
different types of edge modes under periodic driving. Finally, we link the
presence of edge modes to a steady state current.
