Is profanity illegal? Whether in public or private?

You can say whatever you want under the First Amendment, but just because you can say whatever you want, speaking your mind with the use of profanity CAN have legal consequences. If you run afoul of obscenity laws, or the statement lacks literary, artistic, political or scientific value, and appeals only to prurient interests, it certainly can be censored. In addition, if you are yelling profanities in the presence of minors, or are using profanities that incite someone to immediate violence, you can certainly be criminally punished. It isn't that the words themselves are illegal, but how you use them may be. It has been said that profanity is the effort of a feeble brain to express itself forcibly. Maybe there's a better way to say what you need to...

To thoroughly answer your question would take some time and research. Of course, sometimes speech (loud speaking, loud music, or other noises may indeed violate statues dealing with disturbing the peace, etc. An American has the right to the peaceful enjoyment of his home and property, for example. If you want a thorough answer, do some legal research or better yet, hire a lawyer. Let me give you some excerpts from a United States S.Ct. case which is still good law and which I have at hand without going to the law library : [Disorderly conduct] In Gooding v. Wilson, 405 U.S. 518, 92 S. Ct. 1103, 31 L. Ed. 2d 408, a defendant was found guilty of using opprobrious words and abusive language in violation of a Georgia statute. The Fifth Circuit Court of Appeals declared the statute unconstitutionally vague and broad and set aside defendant’s conviction. The Supreme Court stated: “The constitutional guarantees of freedom of speech forbid the States to punish the use of words or language not within "narrowly limited classes of speech. Chaplinsky v. New Hampshire, 315 U.S. 568, 571 (1942). Even as to such a class, however, because ‘the line between speech unconditionally guaranteed and speech which may legitimately be regulated, suppressed, or punished is finely drawn,’ Speiser v. Randall, 357 U.S. 513, 525 (1958), ‘in every case the power to regulate must be so exercised as not, in attaining a permissible end, unduly to infringe the protected freedom,’ Cantwell v. Connecticut, 310 U.S. 296, 304 (1940). Government may pass laws which punish ‘fighting words.’ In Chaplinsky, we sustained a conviction under Chapter 378, § 2, of the Public Laws of New Hampshire, which provided: ‘No person shall address any offensive, derisive or annoying word to any other person who is lawfully in any street or other public place, nor call him by any offensive or derisive name . . . .’ Chaplinsky was convicted for addressing to another on a public sidewalk the words, ‘You are a _ _ _ damned racketeer,’ and ‘a damned Fascist and the whole government of Rochester are Fascists or agents of Fascists.’ Chaplinsky challenged the constitutionality of the statute as inhibiting freedom of expression because it was vague and indefinite. - The Supreme Court of New Hampshire, however, ‘long before the words for which Chaplinsky was convicted,’ sharply limited the statutory language ‘offensive, derisive or annoying word’ to ‘fighting" words': "No words were forbidden except such as have a direct tendency to cause acts of violence by the person to whom, individually, the remark is addressed. . . . “The test is what men of common intelligence would understand would be words likely to cause an average addressee to fight. . . . Derisive and annoying words can be taken as coming within the purview of the statute . . . only when they have this characteristic of plainly tending to excite the addressee to a breach of the peace.... “The dictionary definitions of ‘opprobrious’ and ‘abusive’ give them greater reach than "fighting" words. Webster's Third New International Dictionary (1961) defined ‘opprobrious’ as ‘conveying or intended to convey disgrace,’ and ‘abusive’ as including ‘harsh insulting language.’ Georgia appellate decisions have construed § 26-6303 to apply to utterances that, although within these definitions, are not ‘fighting’ words as Chaplinsky defines them.”