In this talk, I will focus on the wavefunction features of the Majorana zero modes in topological superconductors, that have received immense attention in current day research. Recent works with my collaborators show how the oscillations appearing in the Majorana wavefunctions have drastic effects in non-equilibrium dynamics and disordered systems. Some of these results reveal novel features in the phase diagram of the Kitaev chain unexplored before and they also relate to few exact results in the spin chain physics. In the end, I hope to talk about some of our recent work on novel time-decay and relativistic effects in quantum Hall systems in a saddle potential, that relate to features in black hole physics. I will also indicate some future directions that could be pursued in these topics.