Basic Rules, pg. 307:
Underdark Cities. The dark elves build fantastic cities in enormous caverns where food and water are abundant…
etc.
I realize we don't have settings books for Dragonlance, Greyhawk, etc. yet, but some published books do reference them (e.g. the Sword Coast Adventurer's Guide has a section titled "Class Options in Other Worlds"). On those other worlds do Drow reside in the Underdark? Does the Underdark even exist?
Since SCAG (and maybe other resources) seems to provide canonical answers for hypothetical questions such as "is Bladesinging appropriate to use with Qualinesti elves?" and since the Underdark is specifically mentioned in the Basic Rules I'm hoping there can be a canonical answer to this question. Material from past editions is fine if none exists for 5e, but consistency among those materials (i.e., no differentiation between editions) would then be important.
Best Answer
Greyhawk: Descent into the Depths of the Earth
The Underdark concept originated in Greyhawk with the in the AD&D adventure module Descent into the Depths of the Earth, published first as two different modules in 1978, and republished in a compilation module in 1981. See Korvin's answer for more details about this one.
Exandria
Matthew Mercer's world Exandria features an underdark, and is featured a few times in the Critical Role podcast. For canonical material, we see in Explorer's Guide to Wildemount that the Kryn Dynasty lived for a time in Wildemount's underdark before rebuilding their capital city:
And also:
Eberron
Eberron calls their underdark "Khyber", and it is featured in a section of Eberron: Rising from the Last War: