Is the word "all" singular or plural?

I had originally provided a comment to an answer (which is now deleted) saying that all can be either singular or plural. But that's not really correct. As a comment, I didn't have space to explain it properly.

All is used along with a subject that is either singular or plural.

I ate (all / most / some / part) of the cake.

Sometimes, all is used in a sentence with a subject that's considered singular, and sometimes in a sentence with a subject that's considered plural.

That is all.
All of the cars are white.

According to Merriam-Webster, all can be an adjective, adverb, or pronoun, and it can be used in sentences that involve singular nouns or plural nouns.

It can also be a noun itself. When it is, it's considered to be singular.

Whether it's used in a sentence with a singular subject or a plural subject is contextual.

All is well.

Here, it's used in a singular construction. It's being used in the same sense as everything would be:

Everything is well.

On the other hand, this would also be acceptable, if not common:

All (of them) are well.

In this case (assuming of them is not actually present in the sentence), all is being used as a pronoun in place of something like:

Things are well.
People are well.

Or, depending on how you look at it, it's not a pronoun but an adjective for something that simply isn't present:

All things are well.
All people are well.

All hail Caesar!

The verb form really has nothing to do with a singular or plural subject in this sentence.

What this sentence is actually saying is:

All (of you should / must) hail Caesar!

While the sentence is implicitly talking about the plural you, hail is the conjugation used for both the singular and plural form of you.

When all is used with a plural noun, it means every, and the verb agrees: All the cities were represented at the meeting. When all is used with a singular noun, with or without of, it means entire and takes a singular verb: All the city was in mourning.

Is the word "all" singular or plural?  No, it isn't.  The word "all" does not have a grammatical number, which means that it does not conflict with either a singular or a plural referent. 

All is well.

In this sentence, the referent of "all" is singular.  It agrees with the form "is".  The only thing that marks grammatical number in that sentence is the verb form, so we must assume that "all" has a singular referent in this sentence. 

All are well.

In this sentence, the only thing that marks grammatical number is the verb form "are".  We must assume that "all" has a plural referent here.  Only a plural referent would agree with "are". 

If the question is "how is everything?", one sensible answer is "all is well".  In this case, "all" refers to everything, which is a singular idea.  If the question question is "how is your family?", one sensible answer is "all are well".  Here, "all" refers to the members of the family, which is a plural idea. 

The totality is good.  The individuals are good.  The word "all" might refer to either.