  Motivated by the need to understand and simulate the ubiquitous
experimentally-observed zero-bias conductance peaks in
superconductor-semiconductor hybrid structures, we theoretically investigate
the tunneling conductance spectra in one-dimensional nanowires in proximity to
superconductors in a systematic manner taking into account several different
physical mechanisms producing zero-bias conductance peaks. The mechanisms we
consider are the presence of quantum dots, inhomogeneous potential, random
disorder in the chemical potential, random fluctuations in the superconducting
gap, and in the effective $g$ factor with the self-energy renormalization
induced by the parent superconductor in both short ($L\sim1~\mu$m) and long
nanowires ($L\sim3~\mu$m). We classify all foregoing theoretical results for
zero-bias conductance peaks into three types: the good, the bad, and the ugly,
according to the physical mechanisms producing the zero-bias peaks and their
topological properties. We find that, although the topological Majorana zero
modes are immune to weak disorder, strong disorder (ugly) completely suppresses
topological superconductivity and generically leads to trivial zero bias peaks.
Compared qualitatively with the extensive existing experimental results in the
superconductor-semiconductor nanowire structures, we conclude that most current
experiments are likely exploring trivial zero-bias peaks in the ugly situation
dominated by strong disorder. We also study the nonlocal end-to-end correlation
measurement in both the short and long wires, and point out the limitation of
the nonlocal correlation in ascertaining topological properties particularly
when applied to short wires. The goal of the work is to establish with a very
high confidence level the real physical possibility that essentially all
experimentally observed zero bias peaks in Majorana nanowires are most likely
ugly.
