Use “woman” as a noun, and “female” as an adjective. Don’t use “lady” unless you’d use “gentleman” for a man in the same context. (“This drug may cause beard growth in women.” “She will be the first female president.” “A lady never tells.”)
Referring to someone as “a female something” is fine, but referring to someone just as a “female” is depersonalizing. In everyday speech it tends to have a derogatory sound: “He arrived with some female or other in tow.” In medical writing it’s not rude, but it has a jargony sound: “Our study showed that 38% of females experienced . . .” In some contexts, perhaps if you’re referring to women of all ages, you might choose to use “females” instead of writing something like “female infants, girls, and women,” but wherever possible, I would stick to “women,” “girls,” etc.
I can’t say that using “woman” as an adjective (“Stress fractures are more common in women runners”) is wrong, because I see good writers doing it all the time, but find it unesthetic. AP style is to use female as the adjective and woman as the noun. Maybe people are aware of the negative connotations of using “female” as a noun, and overcorrect by not using the word at all. Don’t worry, it’s OK to say that someone is female (if they identify as female; see the GLAAD Media Reference Guide and recent editions of the AP stylebook for some notes on writing about transgender people).
“Lady” for “woman” is . . . unnecessary? patronizing? Perhaps AP says it best: “Lady may be used when it is a courtesy title or when a specific reference to fine manners is appropriate without patronizing overtones” (Associated Press Stylebook 2013).
Also worth mentioning is another guideline from AP: use “girl” only up until the eighteenth birthday. For adults, use “woman” or “young woman.”
