The gifts and their duration
The spiritual gifts and their duration (1 Corinthians 13:8-13) — continuing in the church or ceased with the apostolic foundation; the Pentecostal, charismatic and cessationist readings.
1 Corinthians 12:4-11
4Now there are various kinds of gifts, but the same Spirit.
5There are various kinds of service, and the same Lord.
6There are various kinds of workings, but the same God, who works all things in all.
7But to each one is given the manifestation of the Spirit for the profit of all.
8For to one is given through the Spirit the word of wisdom, and to another the word of knowledge, according to the same Spirit;
9to another faith, by the same Spirit; and to another gifts of healings, by the same Spirit;
10and to another workings of miracles; and to another prophecy; and to another discerning of spirits; to another different kinds of languages; and to another the interpretation of languages.
11But the one and the same Spirit produces all of these, distributing to each one separately as he desires.
What each tradition says
Rabbinic / Midrashic
Majority viewThe prevailing opinion, according to The Jewish Encyclopedia, is that inspiration ceased with the last of the Prophets: Haggai, Zechariah, and Malachi. The source cites Tosef., Soṭah, xiii. 2; Seder 'Olam R. xx.; Sanh. 11a; and I Mace. iv. 46.
The Jewish Encyclopedia (article unsigned), The Jewish Encyclopedia (New York: Funk & Wagnalls, 1901-1906), s.v. 'Inspiration' — section 'Of the Holy Scriptures'
Patristic
Irenaeus reports hearing many brethren in the Church who possess prophetic gifts. Through the Spirit they speak all kinds of languages, bring hidden things of men to light for the general benefit, and declare the mysteries of God.
Catholic
According to Joseph Wilhelm, in the course of time prophecy became less common. It did not, however, ever disappear altogether.
Joseph Wilhelm, The Catholic Encyclopedia, vol. 3, s.v. 'Charismata' (New York: Robert Appleton Company, 1908) — section on prophecy among 'Charismata given for the inner life of the Church'
Orthodox
Chrysostom says that if both these were brought in for the faith, then once faith is sown everywhere, their use is henceforth superfluous. But loving one another shall not cease; rather it will advance further, both here and hereafter, and then more than now.
John Chrysostom (c. 349-407), Homilies on the Epistles of Paul to the Corinthians (trans. Talbot W. Chambers), in Nicene and Post-Nicene Fathers, ser. 1, vol. 12 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1889) — Homily XXXIV, section 2 (on 1 Cor 13:8)
Reformed / Protestant
Warfield reports that the post-Reformation theologians, a very clear-headed body of men, taught with great distinctness that the charismata ceased with the Apostolic age. He presents this as their clear teaching.
Benjamin B. Warfield (1851-1921), Counterfeit Miracles (New York: Charles Scribner's Sons, 1918) — ch. 1, 'The Cessation of the Charismata' (opening pages)
Evangelical
Albert Barnes reads 1 Corinthians 13:8 as saying the gift shall cease to be exercised, be abolished, and come to naught. In the light and glory of the world above there will be no further use for it, so it shall cease; God shall be the teacher there.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (enlarged ed. by Robert Frew; CCEL carries the Baker reprint) — Note on 1 Corinthians 13:8 (the older evangelical reading)
Evangelical
Pym holds that we should desire spiritual gifts, if the Lord sees fit to grant them. He sees at least a great probability that they will be granted before the day of the Lord comes.
William Wollaston Pym (1794-1867), An Inquiry concerning Spiritual Gifts (Hartford: G. & C. Carvill, 1832) — ch. III, conclusion (on the extent and duration of the gifts)