  Full-shell nanowires have been proposed as an alternative nanowire design in
the search of topological superconductivity and Majorana zero modes (MZMs).
They are hybrid nanostructures consisting of a semiconductor core fully covered
by a thin superconductor shell and subject to a magnetic flux. In this work we
critically examine this proposal, finding a very rich spectral phenomenology
that combines the Little-Parks modulation of the parent-gap superconductor with
flux, the presence of flux-dispersing Caroli-de Gennes-Matricon (CdGM) analog
subgap states, and the emergence of MZMs across finite flux intervals that
depend on the transverse wavefunction profile of the charge density in the core
section. Through microscopic simulations and analytical derivations, we study
different regimes for the semiconductor core, ranging from the hollow-core
approximation, to the tubular-core nanowire appropriate for a semiconductor
tube with an insulating core, to the solid-core nanowire. We compute the phase
diagrams for the different models in cylindrical nanowires and find that MZMs
typically coexist with CdGM analogs at zero energy, rendering them gapless.
However, we also find topologically protected parameter regions, or islands,
with gapped MZMs. In this sense, the most promising candidate to obtain
topologically protected MZMs in a full-shell geometry is the nanowire with a
tubular-shaped core. Moving beyond pristine nanowires, we study the effect of
mode mixing perturbations. Strikingly, mode mixing can act like a topological
$p$-wave pairing between particle-hole Bogoliubov partners, and is therefore
able to create new topologically protected MZMs in regions of the phase diagram
that were originally trivial. As a result, the phase diagram is utterly
transformed and exhibits protected MZMs in around half of the parameter space.
