Comment by deaddodo

You can deny entry on a non-discriminatory basis. E.g., a bar can kick out an individual Black American for being a nuisance or otherwise troublesome, they can't kick out a black guy for wearing blue (unless it's a blanket ban and reasonable, such as it being a theme bar) or being black.

This is why the signs are always phrased as "we reserve the right to refuse service to anyone/any person".

As with most things though, this is just the minimum federal regulation and states will handle how far they take it differently. There are jurisdictions that wouldn't touch a "no tattoos" policy with a ten-foot pole at the risk of a lawsuit. While there are others that are more lax.