The Qur'an
Male homosexuality is only implied in the Qur'an, and there is no mention of lesbians or transexuals.
The story of Lot is repeated numerous times (e.g., 26:165-6, 27:55, and 29:28-9).
The Yusuf Ali translation of 26:165 runs: "Of all creatures in the world will ye approach males, and leave those whom God has created to be your mates, Nay ye are a people transgressing all limits"
The Palmer translation of 27:55 is: "And Lot ... said to his people: 'Do ye approach an abominable sin while ye can see? do ye indeed approach men lustfully rather than women? nay! ye are a people who are ignorant.'"
These passages reflect the post-classical Jewish and Christian interpretation of the Sodom narrative, as well as Aristotle's widely accepted (but incorrect) view that animals do not engage in homosexual acts.
In context, the Qur'an mentions other cities which were destroyed, not just Sodom; including the legendary cities of 'Ad and Thamud. These have much different narratives. For instance, in Thamud "there were in the city nine persons who despoiled the land and did not right." (27:49). In Surah 11 a parallel is drawn between the story of Lot and the Biblical flood narrative of Noah. This is a constant theme that runs throughout the Qur'an. It draws freely from Biblical, Talmundic and traditional Arabic lore of civilizations overwhelmed by catastrophes brought on by hubris.
The common motif of these stories is that the people of these cities defy God, and ignore his prophets; not that they engage in particular sexual practices. God is warning, through the Qur'an, that He is the creator and destroyer of all things. This is a much larger concern, on a cosmic scale, than what people do in their bedrooms.
There is a possible mention of male homosexuality in Surah 4:16. Yusuf Ali translates this as:. "If two men among you are guilty of lewdness, punish them both. If they repent and amend, Leave them alone; for God is oft-returning, Most Merciful" (emphasis inserted). Palmer's translation of the same passage is: "And if two of you commit [adultery], then hurt them both; but if they turn again and amend, leave them alone, verily God is easily turned, compassionate". (Adultery is implied from the previous paragraph). Palmer notes: "the commentators are not agreed as to the nature of the offence here referred to. The punishment to be inflicted is also the subject of dispute." This stands out here, because this Surah (The Women) codifies a number of laws and regulations about sexual behavior, and in each case except for this, the text lays out specific punishments.
There is also a cryptic passage in Surah 76: one of the rewards in Paradise is described as "eternal boys...[like] scattered pearls...and when thou seest them thou shalt see pleasure and a great estate." (Palmer). Whether these are supposed to simply be attendants or companions is left to the imagination.
There is, however, explicit condemnation of homosexuality in the Hadith, which are traditional sayings from early Islam which have acquired legal status. For instance, Williams in his anthology Islam, quotes the following Hadith (p. 83): Bukhārī . . . from Ibn ‘Abbās: "The Prophet cursed men who act like women and women who act like men, and said, 'Drive them from your houses.' He expelled such people, and ‘Umar did it as well."
Islamic societies through history have both tolerated and persecuted LGBT people, sometimes at the same time. However, there is very little in the core text of Islam, the Qur'an, which support the harsh punishments and ostracism which gays are subjected to in contemporary Islamic society, and the relevant passages are either vague or tangential.
