I am interested to know what are Islam views on the nature of individual souls.

The soul is the real self of a human being, residing in his or her body during life on earth. We don’t know the exact nature of it. We know it has no gender or race, and it has been honored by God. Interestingly, it departs the body temporarily, while we are asleep. Perhaps this could explain dreams, and especially the true ones. The soul enters the body during early pregnancy stages and exits, finally at the time of death.

وَيَسْأَلُونَكَ عَنِ الرُّوحِ قُلِ الرُّوحُ مِنْ أَمْرِ رَبِّي وَمَا أُوتِيتُم مِّنَ الْعِلْمِ إِلَّا قَلِيلًا

AND THEY will ask thee about [the nature of] the soul. Say: "The soul is one of the commands of my Lord; and you have been granted very little of knowledge."