Can you sue someone for posting and displaying offensive signs in public?

Mere spoken or written words of general abuse, however rude, annoying, ill-natured or despicable do not constitute slander or libel (defamation) unless a plaintiff shows special damages.  Under New York law, only false assertions of facts, not expressions of pure opinion, regardless of how unreasonable or offensive they may be, can be grounds for defamation.  Expressions of opinions are protected under the U.S. Constitution.  However, where the opinion implies that it is based on facts but is unknown to those hearing or reading it, it is a "mixed opinion" and may be grounds for defamation.  The issue of whether a statement is a pure or mixed opinion is a question for the courts to determine. 

Here, your statement about the hairdresser is an expression of pure opinion.  While the words may be vexatious and discourteous, they are not defamatory and receive federal constitutional protection.