  Helicity-dependent incoherent pion photoproduction in the reaction \vec
\gamma \vec d\to \pi NN is studied in the framework of the diagrammatic
approach. Contributions to the reaction amplitude from diagrams corresponding
to impulse approximation as well as NN and \pi N interactions in the final
state have been evaluated. The elementary \gamma N\to \pi N operator is taken
from the MAID and SAID models. A detailed comparison of the predictions with
recent experimental data by the GDH and A2 collaborations at energies below 500
MeV is presented. Reasonable agreement with the data on the yields and cross
sections for polarized beam and polarized target has been achieved in all
channels. The unpolarized data of the GDH and A2 collaborations have also been
analyzed within the approach. A strong overestimation for the neutral channel
has been found. At the same time, the model provides a quite satisfactory
description of the unpolarized data for the charged channels. The sensitivity
of the obtained results to the choice of the elementary \gamma N\to \pi N
operator is discussed in detail. The contribution of the \gamma d\to \pi NN
reaction to the GDH sum rule for the deuteron up to a photon energy of 1.65 GeV
has been evaluated with the result of 235\pm 25 \mub.
