I know the grammatical difference between good (an adjective) and well (an adverb). However, I get the feeling that people switch between saying e.g. "I hope you are doing good" and "I hope you are doing well", both with the underlying message of "I hope you are fine". Above that, people sometimes say "I hope you are good"! I get the impression that it's a local or dialectal difference.

As a non-English native speaker I learnt in school to say "I am fine, thanks. How are you?"

My question is: what is the difference between these expressions? Which one is more appropriate?

Best regards,
AnonymousWhich one is more appropriate?
It depends on the situation. It's difficult to think of all the possible situations you might want to know about, so let's restrict it to just a few. Note that it is a common mistake to substitute good for well.

When you're asked how you are:

I'm fine. -- standard answer; standard formula
I'm well. -- I'm in good health.
I'm good. -- not appropriate

With I hope you ... :

I hope you are fine. -- not used much
I hope you are well. -- I hope you are in good health.
I hope you are good. -- not appropriate

I hope you're doing fine. -- not used much
I hope you're doing well. -- I hope everything is going well with you.
I hope everything in your life is satisfactory.
I hope you're doing good. -- not appropriate.

In general:

to be good -- to act ethically
to be good at something -- to be skillful at doing it
to be good for something -- to be appropriate or useful for doing something
to be good for someone -- to be helpful or useful to that person
to do good -- to do charitable work; to contribute to charities

to be well -- to be in good health; not to be sick
to do well -- to have an easy life; to make enough money; to succeed

to do something well -- to do it with excelence

Based on my understanding, "I hope you are doing well" is usually used to tell a person that the speaker hopes the person has everything going on smoothly. "I hope you are doing good" is, as far as I'm concerned, wrong. "I hope you are fine" is usually used by the speaker to tell a person who is usually mentally or physically hurt. For example, you approach a sick patient and you tell him: "I hope you are fine" rather than "I hope you are doing well".

Anyway I'm not a native speaker. I'll like to read their opinions too.
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 CalifJim's reply was promoted to an answer.
I'll add to CJ's list only the common 'I'm good'' as a response to an inquiry after need:

A: Do you want another beer?
B: No, I'm good.

We must also differentiate the same phrase, I think, as this sounds quite common and natural to me:

A: How's your health? How've you been?

B: Oh, good, good. (Or 'fine, fine', but not 'well, well')

'Good' and 'fine' then are adjectives modifying 'health'.
It is OK to say, "I'm good," people seem to forget is that it's standard to use adjectives—such as "good"—after linking verbs. When you do it, they are called predicate adjectives, and they refer to the noun before the linking verb. That's why, even though "good" is primarily an adjective, it is OK to say, "I am good": am is a linking verb, and you can use adjectives after linking verbs.

Aside from the linking-verb-action-verb trickiness, another reason people get confused about this topic is that "well" can be both an adverb and a predicate adjective. In the sentence "He swam well," "well" is an adverb that describes how he swam. But when you say, “I am well,” you're using "well" as a predicate adjective. That's fine, but most sources say "well" is reserved to mean “healthy” when it's used in this way. (1, 3, 4) So if you are recovering from a long illness and someone is inquiring about your health, it's appropriate to say, “I am well,” but if you're just describing yourself on a generally good day and nobody's asking specifically about your health, a more appropriate response is, “I am good.” - See more at: http://www.quickanddirtytips.com/education/grammar/good-versus-well?page=1#sthash.Hdc8G2QK.dpuf
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If I say I hope you are good, is that proper