  We establish the sharp conditions for the embedding between Wiener amalgam
spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces
$L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which
partially improve and extend the main result obtained by Guo et al. in J.
Funct. Anal., 273(1):404-443, 2017. In addition, we give the full
characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and
$\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$. Especially, in the case of
$\alpha=0$ with $M_{p,q}^{s,\alpha} = M_{p,q}^s$, we give the sharp conditions
of the most general case of these embedding. When $0<p\leqslant 1$, we also
establish the sharp embedding between Wiener amalgam spaces and Triebel spaces
$F_{p,r}^{s}$.
