What's the difference in use of 女, 女人 and 女子? Are those roughly equivalent? Do they belong to the same register of language?

According to mdbg.net:

女 (nǚ​) – female / woman / daughter
女人 (nǚ​rén​) – woman
女子 (nǚ​zǐ) – woman / female
There are detailed answer from the following thread at CantoDict forums on the difference between 女人 and 女子

女子 Female - Used in the most formal situations, such as news reporting, e.g., 一名女子 (a female), or 女子組 (the female group as in Olympic sports events), or 小女子 (how a woman addresses herself in historical drama)
女人 Woman - Used in casual situations, such as daily conversation, e.g., 一個女人 (a woman) or 三個女人一個墟 (three women [make as much noise as] a farmer's market)

As for the character 女, it can be a noun or an adjective for &quot;female&quot;.
The only example of 女 as a noun, is with a pronoun, for example 此 (this) as in 此女 (this woman) - a simplified form of 此女子 (this woman).
It is only used in classical writing style.