I can answer for medieval Mogadishu, and maybe the other Somali city-states of Brava, Merca, Zeyla, and others.

Mogadishu was by far the biggest and most powerful Somali city-state, both politically and economically, and militarily as well. The city itself was heavily mercentile, and depended on Somali nomadic clans who inhabited the deserts of the Horn of Africa who brought their camel caravans into the city, wherein they and the city citizenry would barter with Chinese, Catalan, Jewish, Malay, Swahili, Persian, and Arab visitors and traders for foreign-sourced such as silks, Chinese ceramics, gold, ivory, guns, and the likes in exchange for fine horses, cloths, meat, grains, and other goods the Somalis and local citizenry made and goods from the neighboring Abyssinians via frontier trading towns like Harar, as well as goods obtained from northern trading towns and cities like Berbera, Zeila, and Hafun and the like. The city had a big harbour, and was usually filled with ships from China, India, Arabia, the Swahili and Malay states, Egypt, and even Europe. Because of this, Mogadishu was the center of mercentile and maritime traffic on the eastern Indian ocean.

The city itself was described by various Islamic and European accounts as 'exceedingly large' and as full of multistoried buildings and palaces, and very affluent. It was divided into multiple neighborhoods/quarters in which different tribes/clans and ethnic groups each occupied (yes, the city was multiethnic, which I will explain shortly). For example, there was the Xamarweyne quarter, the Shinganei quarter, and others of which I am not aware. The architecture was homogenous, with coral and limestone being the main building materials. Hence, the city had a white/grey appreance, similar to the other Somali city-states of Brava and Merca and Zeila. The Somali city-states were described as 'moorish' in style by Portuguese accounts, and are accurately described a Somalo-Islamic, mixing the megalithic Cushitic Somali architectural style which relied on limestone and stone, with the Islamic architectural styles of the middle east. Homes were, on average, 3-4 stories tall with balconies on upper floors, with very good views of the Indian Ocean for the most part. The neighborhood streets are narrow, with multistoried homes and buildings on both sides. Palaces were large and located in the center. Not alot of minarets, and the skyline was mostly dominated by big, multistoried buildings with a minaret here and there.

The citizenry of Mogadishu was mostly described as 'white' whereas the aristocracy and nobility were ethnic Somalis as described by various travelers and visitors. The reason that the majority of Mogadishu was 'white' was the result of a very large influx of Persian, Turkish, Andalusian/Spanish, and Arab families beginning shortly after the advent of Islam and onwards uninterrupted until the 14th century or so. Prior to this influx, the city was probably a smaller trading town identified as 'Sarapion' by a Greek document from the 1st century AD, and was inhabited by Somalis. With the rise of Islam and the later establishment of the Muslim caliphates, the city, along with neighboring Somali city-states of Brava and Merca and also to a lesser extent the Swahili towns and cities to the south, large numbers of Muslim families began to immigrate to the city, either being attracted to the commercial reputation and economic strength of the city, as was the case for most Arab families and many Turkish ones, or for political and religous reasons, as was the case for the Andalusian/Spanish, Persian, and some Arabs. With the influx of these immigrants, Somalis were quickly reduced to a minority within the city, and thus they firmly confined themselves within the nobility and rulership. Because of this, you could probably tell that the city was very multilingual, with Arabic, Persian, Turkish, and Andalusian being spoken, with a vulgar, heavily Arabized/Persianed/Turkified Somali dialect (ancestral to the 'Benadiri' dialect) serving as a lingua franca. The Somali Sultan was described as speaking both Somali and Arab with equal fluency by a visitor in the 13th century.

An interesting point I want to bring up again is how every neighborhood/quarter in the city was occupied by a different tribe/clan and ethnic group. The Arabs and Persians and Turks in the city came in such large numbers that they began to form clans and tribes, with the Arabs actually bringing their tribal groupings to regroup within the city. To this day, the Shinganei quarter is so named because it was occupied by the Shanshiyah tribe of Persians who named themselves after the Shanshiyah region of Persia which they hailed from, with similar quarters being so named after Arab tribes, or after Persian or Andalusian place names, alluding to the ethnic group/tribe which occupied the quarter/neighborhood formerly.

The city was governed by a King/Sultan with a council of his councilmen who hailed from all quarters of the city, representing the myriad groups inhabiting the various Arab, Persian, Turkish and Andalusian enclaves. The city had a standing army of its own, and according to Chinese sources, trained and focused heavily on archery. The Sultan/King was a powerful figure, and hailed from whichever Somali nomadic clan came to be dominant outside in the neighboring deserts. Hence, whilst the Sultan/King and his family and the rest of the Somali aristocracy live within the city in their own palaces, having adopted a settled and gilded lifestyle, their power relied on their nomadic desert clansmen and kinsmen who herded their camels and horses outside in the deserts.

Social class within the city was defined and seen through the traditional Somali xenophobic lens and traditional Somali social laws, wherein the 5 Somalis noble clans (to which virtually all Somalis belonged) formed a first exclusive social tier, with non-Somali free Muslims (generally without regard to race) forming a second tier, and then non-Muslims forming the last tier. The status of slaves were generally ambigous, and depended on 'race', with Abyssinian/Cushitic 'red' slaves being legally near-equals to ethnic Somali, since the Blood money (diya') of a Abyssinian/Cushitic slave was equal to that of a Somali, being 100 camels. However, non-Abyssinian/Cushitic slaves, especially 'black' ones - without regard to religion - were on the lowest rung of the social order. Non-Muslims were always below the status of a Muslim and also Abyssinian/Cushitic slaves, but since the city relied so heavily on trade, tolerance was most likely displayed and prevailed. However, the cities prejudicial and racist charecter was still out in the open, having earned the wrath of the Chinese chronicler Fie Xiu, who would write on his visit to the city, in which he described the citizenry of Mogadishu as mean and full of prejudice, and stupid.

Oh, and having mentioned slaves. Miles within the city, an extensive plantation system was formed by the nomadic Somali clansmen beginning in the 11-12th centuries which was wholly reliant and dependent on mass-slavery and forced human labour and expliotation. Mogadishu supplied the slaves for this behemoth of a agricultural industry by importing tens of thousands of 'black' Bantu peoples from the Swahili coast, and it later came to be a prime slave importing city, and the slave populace within the city shot up within the 12th century and stayed the same proportionally till the 17th. Many of the Arab, Persian, Turkish, and Andalusian families of the city had household slaves, with the Somali aristocracy forming a rank and file of 'Mamluke' slave-soldiers, although some of these (if not most?) Mamluke slave-soldiers are reported as non-black and they seem to have had diverse origins. These slave-soldiers were maintained by the Ajuran subclan and all other Hawiye clansmen that later came to rule the city, with the Ajuran Hawiye 'House of Garen' (who ruled Mogadishu and annexed it into their empire during the 11-12th century) maintaining them as a means of protection and ensuring their own safety against their own non-Ajuran Hawiye clansmen or even other rival Somali clans since these slave-soldiers had no blood loyalties and clan-affiliation and had no worth or status within the Somali clan system as they weren't Somalis to begin with.

Also, with the advent of Ajuran rule in Mogadishu, Mogadishu also had a navy which formed apart of the larger Ajuran navy. It was big enough to make the Portuguese Indian Ocean fleet opt to not attack Mogadishu, and later engage in a seaborne invasion of the Swahili coast with the rest of the Ajuran navy and their defeated Omani allies and drive the Portuguese from the Swahili coast and engage in intermittent naval engagements with Portuguese ships.

Can you be a little bit more specific? The Muslim world by the late Middle Ages was reaching its modern expanse, and spanned from West Africa to Indonesia. As you can imagine, your "typical" Muslim city would be very different based on where you were in the world and its society would be more dependent on the local culture and geography rather than on purely religious grounds.