As this is GQ, I’ll refrain from taking umbrage. Muslims pray five times a day. The prayer, known as nam`az, consists of excerpts of the Koran and prayers of thanks to and rememberence of God. It is not sung. Rather, it is recited, often melodiously, but it is decidedly not sung.
The distinction here, I feel between the use of “sung” and “recited”, is that “singing” implies music, and music is considered inappropriate for a mosque.
To get back to the OP, there is quite a lot of Islamic religious music. As has been mentioned, the qawalis, which are a mainly Shia Islamic tradition, that originated in the Indian subcontinent, from I believe, the Hindu tradition of garbis, and Hindi religious music. As music is considered inappropriate for within the mosque, many qawalis are sung in the holy areas outside a mosque, or near/at monuments to the Shia Imams. For the best example of these, see the qawalis of the late Nusarat Fate Ali Khan. The qawalis tend to use imagery of the complete and utter devotion (ishq) of a beloved to his/her lover as a symbol for the bond between man and God.
Likewise, although strictly not music (see above comment about ‘music’ being inappropriate in a mosque, there is also the tradition of ginans and qasidas.
The ginans are Shia Ismaili devotional poems which can be recited in the mosque. Again, they are very melodious, have set patterns of recitiation (or ‘tunes’ for want of a better word), and the ginans at least, are again based very much on Hindi garbis, and written in old Hindi or local Indian dialects. In fact, many ginans contain references to Indian (Hindi) mythology, such as from the Mahabarat and the Ramayan. These ginans are prominent amongst those communities which converted to Islam from Hinduism on the Indian sub-continent.
Qasidas are an Iranian/Afghani/Persian/Central Asian version of ginans. Like ginans, they have set ‘tunes’, and they are mainly recited in Farsi or Pashtu.
With some googling, you should be able to bring up audio files of examples of many of these.