Running a 5e campaign, I know a lot has changed about Menzo. For instance Lolth is no longer the sole power with her children coming back. My campaign is taking us to Menzoberranzan, but I don't really know how to describe it, or shops, or houses there. I've been googling and searching all morning but I can't find more than the wikis. The wikis are great but they describe history and events, not physical descriptions of architecture or the common goings on.
I'd highly recommend that you purchase or borrow Out of the Abyss, since it is the only source of information about Menzoberranzan in 5th Edition of Dungeons & Dragons. The information is pretty huge and is about 20 pages long. There are lots of physical descriptions. Furthermore there is tons of material about the Underdark. If your adventure is taking place there, it would help you a lot.
Not to make you aggrieved, here is some crucial info:
- Population: 20 000 drow plus thousands of slaves (of various races).
- The government is a Matriarchal theocracy worshiping Lolth, the Demon Queen of Spiders.
- The city has a large standing army of trained drow warriors and mages, bolstered by armed slaves and magical wards. The citizens themselves form a formidable militia.
- You can easily purchase well-trained slaves, fungi, molds and exotic creatures for food (even ones imported from the surface), poisons, potions, oils and elixirs, jewelry, perfumes and silk.
- The most important organizations are:
- The ruling Council (comprised of the matron mothers of the eight most powerful drow noble houses in the city)
- the Church of Lolth (based in Arach-Tinilith)
- Breagan D'aerthe (company of drow spies, mercenaries and assassins)
There is an early R. A. Salvatore Drizzt book set in Menzoberranzan: Homeland, the first book in the Dark Elf Trilogy. I seem to remember Drow using levitate abilities to gain access to buildings, but that was a long time ago, so I could be way off.
As I recall, the city is dominated by powerful houses which war with each other. I imagine the holds of the houses being build up in and upon massive stalagmites which support the cavern which houses the city, with hubs of housing and structures surrounding the base. The slums would be the areas furthest away from these seats of power.
For strict 5E material, you should stick to Momonga-sama's answer .
Chapter 4 in this supplement is devoted to the City of Spiders, spanning around 20 pages. This chapter includes tons of descriptions and a handful of illustrations (other than the ones in Lino Frank Ciaralli's answer).
Now, I mentioned this isn't quite 5E because it was published while 4E was the norm. And while this supplement is intended to be edition-neutral, it covers the period from -3917 DR to 1480 DR. Which means all of this happens before The Sundering.
On a final note, I've also found people recommending the 4E Underdark supplement (PDF). I don't own this one and I haven't read it, but it could be useful. (Thanks to @keithcurtis for pointing out that this supplement does not take place in the Forgotten Realms, nor is there any mention of Menzoberranzan)
Great detail so far. If I might add to it from even earlier sources:
In 2nd Ed there was a whole boxed set: Menzoberranzan - the Famed City of the Drow that is chocker block full of pictures and descriptions of the city.
For some visuals there was a video game that had some nice (for the time) graphics of both interior and exterior scenes in the city.
I would recommend the book series War of the Spider Queen. It contains much information about the different parts of the city, and special attributes about different areas within the city (like how the market area has spells of silence around it to prevent the noise from bothering the other parts of the city. That series also gives a detailed understanding of the non-drow races living there. It was the book series that got me into DnD in fact.
Sword Coast Adventurer's Guide (p. 101) has a little blurb about the City of Spiders.
It is described as a vast cavern, about two miles long. One wall plateau is consumed by the Academy, Tier Breche, where young priestesses, soldiers, and wizards of the noble houses go to be educated. Many huge stalagmites climb toward the ceiling, and these are turned into dwellings, in much the same way surface elves coax trees into a desired shape rather than simply clearing them and building something from scratch.
Walled off in the center is the marketplace, where the Drow and others can conduct a brisk business, but the higher-up Drow don't need to deal with it if they don't wish to.
Rothe herds are raised a little ways off, and I'm sure some river flows nearby, to supply water. Though the Drow can see somewhat in darkness, the region is dimly illuminated by glowing fungus, so others can function as well.