Is "both" singular?

The word both is a determinative (in the terminology of CGEL). Determinatives are not per se singular or plural (except these and those), but they select singular or plural nouns. When both appears without the head noun, as it does here, there is a fusion of functions and it becomes both the determiner and the head of the plural NP. The verb then agrees with the plural NP.

(ignore the clumsy grey line above Head)

It acts only like plural because, unlike each¹ and none², "both" refers to all of the elements and not to one of them.

Read these examples from the OALD:

• Both of us were tired. - and not - **Both of us was tired.*
• Both of them were French - and not - **Both of them was French*.

(1) - Each is usually singular: "each of us knows about you".
(2) - None can be both singular or plural: "none of us is/are going home." (the singular form is considered formal in BrE).

^{Note: for more accurate explanations, visit the links, otherwise if I add too much info, this answer gets off topic.}