That which is vulgar, obscene, or profane (title reflects contents)

Profane refers specifically to obscenities that are showing disrespect to God (or gods). Vulgar is used to describe obscenities that are not profane. However, if you are using obscenities in what should be a sacred setting, you are showing disrespect, and that is profane (even if the word used is simply vulgar).

In other words, all are obscenities, broken down into two groups: profanities and vulgarities.

Vulgar: the basic meaning is low class, not refined.

Profane: the basic meaning is insulting to God or religion, to desecrate

Obscene: the basic meaning is dirty, offensive to modesty, or decency.

Using that word would certainly be vulgar, it is not the sort of word you would use when you visited the queen. It would most likely be obscene in many settings, since decency and modesty require abstaining from referring to sex, especially in a harsh way. Of course in some settings this might be considered complete appropriate, in which case it would not be obscene.

The word is not profane intrinsically since it is not religious in nature (such as "God damn", or "Jesus H. Christ" might be.) However, in a church setting when appropriate modesty and decency might be expected, the occupants of the church might find the language insulting to the sanctified nature of the place, and consequently profane.

FWIW, profanity and swearing in general usually falls into one of three categories in English: sexual references, references to bodily functions, and blasphemous language. Beside this there is a lot of swearing that is basically a minced form of this, such as corblimey, gosh darnit, or my favorite new one: "shut the front door."

I don't think it has to be any particular one of those three. They're all correct. As your example shows, in different contexts it's OK to call a word different things. 'Vulgar' is the word they use in a dictionary for that register; 'profane' is a churchy description of the word; and 'obscene' sounds right for public.

How do you tell for other words? Our dictionaries aren't so detailed that they give every nuance, so it takes living with the examples day in and day out (and it helps a lot to grow up with the language, if you have that choice).

I'd argue that it's none of these things. It is a generic, yet taboo, intensifier that has no intrinsic meaning.

Only one definition of "fuck" has a fundamentally sexual component. Even in that case, it could be replaced with any other nonsense word and retain its meaning, if not its offense.

But in other cases, (What the fuck? Fuck yeah! You stupid fuck. That's fucked up. Fucking brilliant!) the word serves only to intensify the sentence.

Unfortunately, the word has become so offensive in American culture, that its use can result in criminal charges including contempt of court, assault of an officer and incitement to violence. Now that is fucked up.