What I can glean from a couple of books by David Nicolle is that they had a core of askar professional household troops,
sword armed, slave recruited ghulams.
There were a variety of non-professional troops : ajnad provincials and jund soldiers, most of whom were Turks or Kurds plus a few arabs.
The main cities had their ahdath urban militias, which could also be used outside the cities. Not usually considered to be very reliable soldiers, but resolute in defence of their particular city.
The army from Damascus in 1138 also included Armenians.
Later Damascus also had the garrisons of Tyre and Sidon when these fell to the Crusaders, along with other refugees.
Major cities could also call upon allies, in the form of Turcomans from further afield.
Supplementing these were religiously motivated volunteers called mutatawiah.
Another list of Aleppo soldiery included siege engineers from Aleppo and Khurasan, Armenians, professional naffatun, plus a large number of non-combatant labourers.
Generally speaking there was a tendency to source troops from a variety of sources to prevent any particular tribe or nationality being able to take over the military and hence the state.
(Bedouin were usually careful to avoid getting involved in too much fighting between rival power factions.)