  Consider the McKean-Vlasov SDE $$
  dX_t=\langle b(X_t-\cdot),\mu_t\rangle dt+dW_t,\quad
\mu_t=\operatorname{Law}(X_t), $$ where $W$ is the $n$-dimensional Brownian
motion and $b:\mathbb{R}^d\to\mathbb{R}^d$ is a measurable function. First
assuming $b\in L^\infty$, we prove that the law $\mu_t$ of $X_t$ has a density
$p_t$ with respect to the Lebesgue measure, which is continuously
differentiable with gradient being $\gamma$-H\"older continuous for each
$\gamma\in(0,1)$. Assume further that $b\in \mathcal{C}_b^1$, we prove that the
density $p_t$ is infinitely differentiable. In the regularization by noise
perspective, this shows McKean-Vlasov SDEs tend to have a smoother density
function than SDEs without density dependence, under the same regularity
assumption of the coefficients. We observe similar phenomenon for singular
interaction kernels satisfying Krylov's integrability condition, for
distributional kernels $b\in B_{\infty,\infty}^\alpha$, $\alpha\in(-1,0)$, and
for processes driven by an $\alpha$-stable noise for $\alpha\in(1,2)$.
