We define and study wave-front sets for weighted Fourier-Lebesgue spaces when the weights are moderate with respect to associated functions for general sequences $\{ M_p\} $ which satisfy Komatsu's conditions $(M.1) - (M.3)'$. In particular, when $\{ M_p\} $ is the Gevrey sequence $(M_p = p!^s$, $s>1)$ we recover some previously observed results. Furthermore, we consider wave-front sets for modulation spaces in the same setting, and prove the invariance property related to the Fourier-Lebesgue type wave-front sets.