A number of philosophers believes [duplicate]

Formally, is it correct to write:

A number of questions has been asked here.

or:

A number of questions have been asked here.

As a non-native speaker of English, I would prefer the former: the subject seems to be "number", therefore the verb ought to be singular, I'd say. However, the latter seems more common, and therefore I believe that my gut feeling is just plain wrong — but I would really like to have a definite answer.

Moreover, is it the same for "a myriad of", "a plethora of", and so on?

"The number" is singular. "A number", however, is plural, and takes a plural verb. Thus, for both informal and formal usage, the following is correct:

A number of questions have been asked here.

See the usage note not quite halfway down the page at Dictionary.com, or this daily writing tip.

Just looking at the words themselves, you would logically conclude that the former construction is preferable, but you're correct in observing that no one uses them that way, because they're wrong. Technically, yes, the subject is singular, but there are many phrases in English that are used to describe amount, such as "a lot", "a bunch", and "a few", which are singular but always used as though they were plural. It's neither colloquial nor informal to say "a number have been asked".

There is some disagreement on whether "none" should be used as singular or plural, since it was originally a contraction of the word for "not" and that for "one" or "any". Generally speaking, both "none is" and "none are" are correct, but the latter is, again, more common.