There is a new paper in American Economic Journal: Applied Economics by David H. Autor, Alan Manning, and Christopher L. Smith, here is the abstract:

We reassess the effect of minimum wages on US earnings inequality using additional decades of data and an IV strategy that addresses potential biases in prior work. We find that the minimum wage reduces inequality in the lower tail of the wage distribution, though by substantially less than previous estimates, suggesting that rising lower tail inequality after 1980 primarily reflects underlying wage structure changes rather than an unmasking of latent inequality. These wage effects extend to percentiles where the minimum is nominally nonbinding, implying spillovers. We are unable to reject that these spillovers are due to reporting artifacts, however.

Here are earlier, ungated versions of the paper. Overall my read of this is that many people are leaping in too quickly and making unsupported claims about how the minimum wage is connected to income inequality.