The fall of the day star
Whether the taunt of the fallen day star speaks only of Babylon's king or also of the fall of Satan.
Isaiah 14:12-15
12How you have fallen from heaven, shining one, son of the dawn! How you are cut down to the ground, who laid the nations low!
13You said in your heart, “I will ascend into heaven! I will exalt my throne above the stars of God! I will sit on the mountain of assembly, in the far north!
14I will ascend above the heights of the clouds! I will make myself like the Most High!”
15Yet you shall be brought down to Sheol, to the depths of the pit.
What each tradition says
Rabbinic / Midrashic
Kaufmann Kohler states that, taken together with the (Babylonian?) mythology of Lucifer in Isaiah 14:12, it seemed to take for granted the existence of evil spirits. He describes them as working antagonistically to God through the evil practices of witchcraft, astrology, and the like.
Kaufmann Kohler, Angelology (Jewish Encyclopedia, 1901-1906) — Angelology, section 'Fall of the Angels'
Patristic
Gregory of Nyssa states that the evil poison worked its effect against man’s life. Then man, that noble thing and name, the image of God’s nature, was made like vanity, as the prophet says.
Gregory of Nyssa, On the Making of Man (NPNF series II, vol. V, trans. H. A. Wilson) — Ch. XX (the enemy's poison working against man's life)
Catholic
Aquinas takes Isaiah 14:13-14 as spoken in the person of the devil. Augustine says that the devil, inflated with pride, wished to be called God.
Thomas Aquinas, Summa Theologiae, Prima Pars (Fathers of the English Dominican Province trans.) — I q. 63, a. 3, 'On the contrary' (Isa 14:13-14 in the person of the devil)
Orthodox
No sourced position found. Checked CCEL/archive.org Greek fathers (Basil's Hexaemeron scan, Gregory of Nyssa 'On the Making of Man') and archive.org searches for an Isaiah 14 comment in the Eastern tradition; no citable PD statement applying Isa 14:12-15 to the fall of Satan (or restricting it to Babylon's king) was located. Common orthodox gap for technical Western disputes.
Reformed / Protestant
Calvin says the view that this passage refers to Satan comes from ignorance, since the context plainly shows it refers to the king of Babylon. He adds that ignoring context causes such mistakes, and it was gross ignorance to imagine Lucifer was the king of devils or that the prophet gave him that name.
John Calvin, Commentary on Isaiah, Volume 1 (trans. William Pringle; CCEL text) — Commentary on Isaiah 14:12 ('O Lucifer, son of the morning' — read of the king of Babylon, not Satan)
Evangelical
Spurgeon identifies the falling spirit who descends into hell as the one who was once Lucifer, son of the morning. That same figure, he says, has now become the Father of nights, the Lord of Darkness, Satan, the Fallen one.
C. H. Spurgeon, The New Park Street Pulpit, vol. II (1856): Sermon 97, 'Pride and Humility' (Prov. 18:12) — Sermon 97, 'Pride and Humility', point 3 ('the consequence of pride is destruction')