Word choice for salutation in a formal email to your PhD adviser, Is *Dear Boss* appropriate?

In my view, in academia, the word "Boss" carries a large amount of semantic baggage that you probably don't want to invoke. Although you mean it to be polite and respectful, it also has an ironic context which you may not have detected. By saying "Dear Boss" can also imply that in your opinion they are neither dear to you or deserve the respect as a superior. It is safest to avoid such salutations altogether in emails.

As others have indicated this is very culturally dependent and I think you are bringing too much of your south asian culture to a european environment. Do what the locals do and emulate other peers is the safest and wisest approach.

"Dear Professor X" is still probably still appropriate. Even if they are a full-time researcher and aren't teaching, their official position is probably some variant of "Professor." If their job title is not professor, then I would just go with "Dear Dr. X" and accept that it might be a little too formal.

At least in the US, I would pretty much never use "Dear Boss," even in an email sent to my actual boss in industry. It sounds awkward and weirdly removed. If you want something more formal than a first name, stick to "Ms./Mr./Dr. X."

The first thing to do is to ask your supervisor what she wants to be called. Does she want to keep things formal? Does she want to be called by name? Whatever she tells you is the appropriate salutation to use with her. There is no need to second-guess yourself, nor to go spend too much time over-thinking this.

Now, I know that cultural differences are hard to combine. For this reason I'll give you some pointers based on personal experience. In general, the French culture is one of extreme politeness when it comes to communication. The way people go about this is to use formal pronouns and conjugations when talking to each other. In other words, you'd address your interlocutor as second person plural vous. This is all true until one of the two, usually the one's that higher up in the hierarchy/respect scale (your manager/your spouse's parents), actually tells you "on peut se/tu peux me tutoyer" which means "we/you can use the informal pronoun tu", as opposed to vouvoyer which is what you were initially doing. Tu is the second person singular.

When it comes to titles, the French don't really seem to be fussy about them. Indeed, the main titles used are Monsieur and Madame, i.e. Mr and Mrs (there is also Mademoiselle (Ms) for nubile women, although it's kind of deprecated since it's virtually impossible to safely gauge if one is married or not after they reach adulthood). Medical Doctors here are referred to as Monsieur/Madame, the Prime Minister is Monsieur Valls, the President de la République is Monsieur Hollande.

Therefore, regardless of whether your supervisor is or is not a professor, has or doesn't have teaching assignments, the formal way of addressing her would be "Madame Xyz". Which, in your email would become "Chère Madame Xyz," or "Dear Mrs. Xyz". If however, Madame Xyz explicitly told you to use her first name Asd then a "Bonjour Asd," or "Hello Asd," will do.

The title "Boss" is in no way appropriate nor more polite than Madame. I would not use it in this context, nor anywhere else in general.