The psalm titles
Whether the titles are part of the inspired text or later editorial notes, and what their witness to David is.
Psalm 3:1-8
1Yahweh, how my adversaries have increased! Many are those who rise up against me.
2Many there are who say of my soul, “There is no help for him in God.”
3But you, Yahweh, are a shield around me, my glory, and the one who lifts up my head.
4I cry to Yahweh with my voice, and he answers me out of his holy hill.
5I laid myself down and slept. I awakened; for Yahweh sustains me.
6I will not be afraid of tens of thousands of people who have set themselves against me on every side.
7Arise, Yahweh! Save me, my God! For you have struck all of my enemies on the cheek bone. You have broken the teeth of the wicked.
8Salvation belongs to Yahweh. May your blessing be on your people.
What each tradition says
Rabbinic / Midrashic
Hirsch holds that the psalm superscriptions are late additions. The historical value of “unto David” is no greater than other notes pretending to give a psalm’s occasion, and variants in the versions prove these superscriptions are late interpolations reflecting their authors’ views.
Emil G. Hirsch, The Jewish Encyclopedia (1905), art. "Psalms" — On the superscriptions (the section preceding 'Date of Psalter')
Rabbinic / Midrashic
Rashi on Psalms 3:1 reports that the Aggadists expounded many homilies on this matter. Citing Ber. 7a, he says that when the prophet said evil would be raised against him from his own house, his heart quaked, fearing perhaps a slave or a child of an illicit union would rise against him and not have mercy on him.
Rashi, Rashi on Psalms (The Judaica Press complete Tanach with Rashi, translated by A. J. Rosenberg) — On Psalms 3:1
Patristic
Augustine says the words "I slept, and took rest; and rose, for the Lord will take me up" lead him to believe this Psalm is to be understood in the person of Christ. These words, he says, sound more applicable to the Passion and Resurrection of our Lord than to David's flight from his rebellious son.
Augustine of Hippo, Expositions on the Book of Psalms (Nicene and Post-Nicene Fathers, First Series, vol. 8) — Psalm III — the title and verse 1 (the Psalm read in the Person of Christ)
Catholic
Walter Drum states that Catholic scholars have not insisted the author of the Psalms wrote the titles. They have always considered those titles an integral part of Holy Writ.
Walter Drum, The Catholic Encyclopedia (New York: Robert Appleton Company, 1911), art. "Psalms" — Section "Value of the titles"
Orthodox
No sourced position found. Checked CCEL NPNF2-4 (Athanasius: the Prolegomena list the tract to Marcellinus and De Titulis Psalmorum, but neither is translated in the volume), archive.org search for a public-domain English Chrysostom/Theodoret/Euthymius on the Psalms (none — modern translations only), pravoslavieto.com (Philaret's Catechism; Synod of Jerusalem 1899 — no treatment of the psalm titles).
Reformed / Protestant
Matthew Henry treats the title as a key hung at the door to open the psalm, since knowing its occasion helps expound it. He says it was composed, or at least its substance meditated in David’s thought and offered to God, when David fled from Absalom, who conspired to take away not only his crown but his life.
Matthew Henry, Commentary on the Whole Bible, Volume III (Job–Song of Solomon) — On Psalm 3 — the title as the key to the psalm
Evangelical
Barnes says the title may have been prefixed to Psalm 3 by the author himself, as ancient authors sometimes prefixed their names to their own compositions. It is not absolutely certain, however; it may instead have been done by the collector or arranger of the Psalms, indicating the prevalent belief about authorship and under the Spirit of inspiration.
Albert Barnes, Albert Barnes' Notes on the Bible — On Psalm 3 — the title and its authority