  Spin-incoherent Luttinger liquid (SILL) is a different universal class from
the Luttinger liquid.\ This difference results from the spin incoherence of the
system when the thermal energy of the system is higher than the spin excitation
energy.\ We consider one-dimensional spin-$1$ Bose gas in the SILL regime and
investigate its spin-dependent many-body properties.\ In Tonks-Girardeau limit,
we are able to write down the general wave functions in a harmonic trap.\ We
numerically calculate the spin-dependent (spin-plus, minus, and $0$) momentum
distributions in the sector of zero magnetization which allows to demonstrate
the most significant spin-incoherent feature compared to the spinless or
spin-polarized case.\ In contrast to the spinless Bose gas, the momentum
distributions are broadened and in the large momentum limit follow the same
asymptotic $1/p^4$ dependence but with reduced coefficients.\ While the density
matrices and momentum distributions differ between different spin components
for small $N$, at large $N$ they approach each other.\ We show these by
analytic arguments and numerical calculations up to $N$ $=$ $16$.
