We reserve the right to refuse service.
The sign's message is clear and simple, but the truth is that a business can't reserve a wholesale right to refuse service.
As places of public accommodation, private businesses are subject to federal and state anti-discrimination laws. These statutes prohibit discrimination on the basis of race, color, religion, national origin, disability, gender and sex. Some also include sexual orientation.
And others, well they outlaw even arbitrary discrimination.
For example, California's Unruh Civil Rights Act prohibits all types of arbitrary discrimination. This includes biases based on physical attributes, political beliefs,
and geographical origin.
Courts also tend not to favor arbitrary discrimination. In the past, judges have used consumer protection, unfair business practice, and tort laws to punish such practices.
None of this means that you absolutely cannot refuse to serve a customer. It simply means that you need a legitimate business reason to do so.