  Let $T_m$ be the $m$-th Calder\'on-Zygmund type singular integral. In the
paper, we consider the boundedness of $T_m$ on the generalized product local
Morrey spaces $LM_{p_1,\varphi_1}^{\{x_0\}}\times
LM_{p_2,\varphi_2}^{\{x_0\}}\times\dots\times LM_{p_m,\varphi_m}^{\{x_0\}}.$
And, the boundedness of the commutators of $T_m$ with local Campanato functions
is obtained, also.
