  Incoherent photoproduction of pions on the deuteron in the first resonance
region is investigated with special emphasis on single-spin asymmetries. For
the elementary pion production operator an effective Lagrangian model which
includes the standard pseudovector Born terms and a resonance contribution from
the $\Delta$(1232)-excitation is used. Single-spin asymmetries, both for
charged and neutral pion photoproduction on the deuteron, are analyzed and
calculated for the first time in the first resonance region. The linear photon
asymmetry $\Sigma$, vector target asymmetry $T_{11}$ and tensor target
asymmetries $T_{20}$, $T_{21}$, and $T_{22}$ for the reaction $d(\gamma,\pi)NN$
are predicted for forthcoming experiments.
