  Generation of circularly-polarized (CP) and linearly-polarized (LP)
$\gamma$-rays via the single-shot interaction of an ultraintense laser pulse
with a spin-polarized counterpropagating ultrarelativistic electron beam has
been investigated in nonlinear Compton scattering in the quantum
radiation-dominated regime. For the process simulation a Monte Carlo method is
developed which employs the electron-spin-resolved probabilities for polarized
photon emissions. We show efficient ways for the transfer of the electron
polarization to the high-energy photon polarization. In particular, multi-GeV
CP (LP) $\gamma$-rays with polarization of up to about 95\% can be generated by
a longitudinally (transversely) spin-polarized electron beam, with a photon
flux at a single shot meeting the requirements of recent proposals for the
vacuum birefringence measurement in ultrastrong laser fields. Such high-energy,
high-brilliance, high-polarization $\gamma$-rays are also beneficial for other
applications in high-energy physics, nuclear physics, and laboratory
astrophysics.
