The $L^k$-Wasserstein distance $\mathbb{W}_k (k\ge 1)$ and the probability distance $\mathbb{W}_ψ$ induced by a concave function $ψ$, are estimated between different diffusion processes with singular coefficients. As applications, the well-posedness, probability distance estimates and the log-Harnack inequality are derived for McKean-Vlasov SDEs with multiplicative distribution dependent noise, where the coefficients are singular in time-space variables and $(\mathbb{W}_k+\mathbb{W}_ψ)$-Lipschitz continuous in the distribution variable. This improves existing results derived in the literature under the $\mathbb{W}_k$-Lipschitz or derivative conditions in the distribution variable.