Who's asking: Various clients
One of my clients, in fact, went so far as to say, "William Goldman uses adverbs. Are you saying he shouldn't?"
My answer to that was that William Goldman can do whatever he wants. He's William Goldman, and he rules.
The rest of us, however, should avoid adverbs when writing fiction. It's part of my larger rule, which most writing instructors call, "Show, don't tell." I find that advice useless, though what I say is probably just as vague: "Trust your reader."
Adverbs are bad because they do the readers' work for them, or compensate for lazy writing. Don't tell me that your character said something "teasingly;" I ought to be able to infer that teasing from the content of the dialogue. If I can't, the dialogue's not right. If someone's shrugging, you don't need to say that she shrugged helplessly or tiredly or carelessly or defensively; we should already know that from the action that preceded the shrug.
Likewise, modifying your adjectives is seldom a good idea. "Very," "extremely," "rather" and "quite" are weasel words, qualifiers that belong in political speeches and newspaper columns, not in fiction. Fiction ought to be the one place where characters say and do exactly what you intend them to say or do, without reservation or apology. (Unless that's the character.)
Do not write to me to defend your use of adverbs. Instead, examine your conscience and your prose, go your way and sin no more.