The procession of the Spirit
'The Spirit proceeds from the Father' — with the Western 'and from the Son' (filioque): the procession of the Spirit as a standing East-West fault line (also John 16:7-15).
John 15:26-27
26“When the Counselor has come, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will testify about me.
27You will also testify, because you have been with me from the beginning.
What each tradition says
Rabbinic / Midrashic
No sourced position found. Checked The Jewish Encyclopedia (1905), s.v. 'Trinity' (art. signed K.) — it treats the dogma's rise and the Jewish rejection of it but never mentions the Spirit's procession; a JE site search for 'Filioque' returns no article, and no rabbinic or midrashic text engages the intra-Christian John 15:26 dispute.
Patristic
Gregory of Nazianzus states that the Spirit, because He proceeds from That Source, is not a creature. Because He is not begotten, He is not the Son; and because He is between the Unbegotten and the Begotten, He is God.
Gregory of Nazianzus (c. 329-390), The Fifth Theological Oration (Oration XXXI, On the Holy Spirit), in Nicene and Post-Nicene Fathers, ser. 2, vol. 7 (trans. Charles Gordon Browne and James Edward Swallow; New York: Christian Literature Publishing Co., 1893) — Oration XXXI, section VIII (on John 15:26)
Patristic
Augustine says there are no intervals of time by which it could be shown or even asked whether the Son was born of the Father first and then the Holy Spirit proceeded from both. Holy Scripture calls Him the Spirit of both.
Augustine of Hippo (354-430), On the Holy Trinity (trans. Arthur West Haddan), in Nicene and Post-Nicene Fathers, ser. 1, vol. 3 (ed. Philip Schaff; Buffalo: Christian Literature Publishing Co., 1887) — Book XV, ch. 26 (The Procession of the Holy Spirit from the Father and from the Son is Apart from Time)
Catholic
Maas writes that the filioque is a theological formula of great dogmatic and historical importance. It expresses the procession of the Holy Ghost from both Father and Son as one Principle, and it was the occasion of the Greek schism.
A. Maas, The Catholic Encyclopedia, vol. 6, s.v. 'Filioque' (New York: Robert Appleton Company, 1909) — Opening of the article (before the sections 'Dogmatic meaning' and 'History')
Orthodox
John of Damascus confesses one Holy Spirit, the Lord and Giver of Life. This Spirit proceeds from the Father and rests in the Son, and is equally adored and glorified with the Father and Son.
John of Damascus (c. 675-c. 749), An Exact Exposition of the Orthodox Faith, in Nicene and Post-Nicene Fathers, ser. 2, vol. 9 (trans. S. D. F. Salmond; Edinburgh: T&T Clark, 1898) — Book I, ch. 8 (Concerning the Holy Trinity)
Reformed / Protestant
The Westminster Confession of Faith states that the Holy Ghost eternally proceeds from the Father and the Son. It presents the Holy Ghost as eternally proceeding from both the Father and the Son.
The Westminster Assembly (1643-1653), The Westminster Confession of Faith (1647, with the American alterations), in Philip Schaff, The Creeds of Christendom, vol. 3 (New York: Harper & Brothers, 1877) — Chapter II, section 3 (Of God, and of the Holy Trinity)
Evangelical
Wesley states that the Spirit proceeds from the Son as well as from the Father. He says this may be fairly argued because the Spirit is called the Spirit of Christ in 1 Peter 1:11, and because he is said to be sent by Christ from the Father, as well as sent by the Father in his name.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755); CCEL's 'Wesley's Notes on the Bible' — Note on John 15:26