According to St. Thomas Aquinas, apart of our soul is specifically attached to our body, but another part is detached from it and, at death, detaches entirely. This is the part that moves on, and the rest of our soul will express itself again once we receive a resurrected body. According to the Catechism, our soul is the form of our body; they form one, single nature, just as Christ’s God nature, human soul and physical body all form one Person.
However, I do find it interesting that, as far as I can tell, the only supernatural or heavenly beings to have ever appeared to people on Earth are God, angels and the Blessed Virgin - all of whom have physical, if spiritualized, bodies. I am unaware of the appearance of any other saints throughout history, since none of them have physical bodies to appear in. Rather, they reside in Heaven with one another, in communion between themselves, the angels, Mary and God. They experience the “spiritual thoughts and feelings” of being in the presence of God in Heaven, such as joy/bliss, peace, purity, and love, as tonyrey elucidated. Also according to the Catechism (I think; it may be in another Church official document), the saints would have no power without God specifically giving it to them. They do not “attain” the power to perform miracles through their intercessory prayer, or to appear - this even applies to the Blessed Virgin. But the primary work of heavenly saints is said prayer, which, being a completely spiritual task, they are able to do.
My remaining question is: what precisely constitutes the behaviors or faculties of our soul and spirit, as distinct in character from our brain?