  Supermassive black holes (SMBHs) are commonly found at the centers of their
host galaxies, but their formation still remains an open question. In light of
the tight correlation between the BH mass and the velocity dispersions of the
bulge component of the host galaxy, a BH-host galaxy coevolution scenario has
been established. Such description however still contains many theoretical
uncertainties, including the puzzels about the formation of BH seeds at high
redshifts and the growth channel fueling these seeds. In this work, we
systematically analyze the signatures of different growth channels on MBH
spins. We show that different growth channels can be partially distinguished
with the magnitudes of MBH spins infered from extreme-mass-ratio-inspirals
detected by the Laser Interferometer Space Antenna. In addition, we propose to
measure the correlation between the directions of MBH spins and their host
galaxy spins, which is possible for extreme mass-ratio inspirals happening in
low-redshift galaxies ($z \le 0.3$). With the inclusion of spin direction
correlation different formation channels shall be significantly better
constrained.
