  It is difficult to unambiguously confirm the existence of Majorana zero modes
(MZMs) due to the absence of smoking-gun signatures in charge transport
measurements. Recent studies suggest that the spin degree of freedom of MZMs
may provide an alternative detection method. We study the spin properties of
the superconducting state in Majorana nanowires and the associated
unconventional Josephson effect with realistic experimental parameters taken
from [Zhang {\it et al.}, Nature 556, 74 (2018)]. For a superconducting thin
film with in-plane polarized spin-triplet pairing, an out-of-plane electric
field can generate a supercurrent perpendicular to both the superconducting
spin polarization and the electric field, so we name this phenomena as
superconducting anomalous Hall effect (ScAHE). In a Majorana nanowire, the
regime with finite polarized spin-triplet pairing almost coincides with the
chiral regime, which includes the topological regime. We further study the
effects of polarized spin-triplet pairing in two types of Josephson junctions.
One dramatic finding is that SOC can induce an anomalous supercurrent at zero
phase difference only in the U-shape junction, a basic ingredient of scalable
topological quantum computation. This can be viewed as a consequence of the
ScAHE. Our work reveals that the spin degree of freedom is indeed helpful for
detecting MZMs.
