When is it appropriate to refer to someone as Dr. [Surname]?

The title Doctor is applied to both medical doctors and to anyone with a PhD in any subject.

So far as I know, most English speaking countries don't legislate who may use the title of Doctor. I believe the situation is different in, for example, Germany. Therefore it is a matter of social convention. I expect the law would be broken if you used the title in an attempt to deceive an employer, patient or customer.

It is notable that consultants in hospitals, who are usually highly qualified, use the title "mister".

It is often a matter of context and personal preference. I hold a Ph.D. and so it's not wrong to refer to me as Dr. Gregory. I only ever introduce myself that way when I am on a university campus in my professional capacity. I occasionally allow others to introduce or address me that way (allow meaning I don't correct them in front of third parties) in other contexts, but I don't do it myself. That's my preference for everyone with a Ph.D. and in fact for most medical doctors too. I don't see why, in a meeting of the neighborhood watch, we all need to be reminded about somebody's educational background. That said, if you constantly call the dentist on your block Dr. Whoever you won't be grammatically wrong, just possibly socially wrong.

If the person is a stranger and you're writing about them, go with Dr. Whoever. If it's a real person, ask them, in case they feel as I do about it. I can't resist telling you what my mother, who also holds a Ph.D., did once. Someone introduced her to someone else, saying "Mr. X, meet Ms. Y. " Mr. X responded "Oh, I don't believe in Ms., I will call you Mrs. Y" to which my mother sweetly smiled and said "Oh, if we're going to be technical you can call me Dr. Y" - as you might imagine Mr. X turned very red and excused himself from the conversation quite quickly.

In the UK only Medical Graduates of any academic level are called Doctors, plus those who holding a PhD. Dentists, Surgeons, Veterinarians and anyone else who preforms surgery are Mr/Mrs. In my experience it is only in the USA that every man and his dog wants to be called Doctor.