The Catholic Church has given 33 people so far the honorific title, 'Doctor of the Church'. The basic requirements are (1) sanctity of life; (2) excellence of teaching; and (3) a surviving body of writings, of permanent and inspiring value, that can be recommended to the faithful generally; there are, however, additional complications, since it is a liturgical title for the Universal Calendar, and has to comply with certain liturgical rules and principles. Those who have received the title so far are as follows, according to date of death (sometimes approximate), with the date they were officially given the title in parentheses after.
368 Hilary of Poitiers (1851)
373 Athanasius (1568)
373 Ephrem the Syrian (1920)
379 Basil of Caesarea (1568)
387 Cyril of Jerusalem (1883)
390 Gregory Nazianzen (1568)
397 Ambrose of Milan (1295)
407 John Chrysostom (1568)
420 Jerome (1295)
430 Augustine (1295)
444 Cyril of Alexandria (1883)
450 Peter Chrysologus (1729)
461 Leo the Great (1754)
604 Gregory the Great (1295)
636 Isidore of Seville (1722)
735 Bede (1899)
749 John Damascene (1883)
1072 Peter Damian (1828)
1109 Anselm (1720)
1153 Bernard of Clairvaux (1830)
1231 Anthony of Padua (1946)
1274 Thomas Aquinas (1568)
1274 Bonaventure (1588)
1280 Albert the Great (1931)
1379 Catherine of Siena (1970)
1582 Teresa of Avila (1970)
1591 John of the Cross (1926)
1597 Peter Canisius (1925)
1619 Lawrence of Brindisi (1959)
1621 Robert Bellarmine (1931)
1622 Francis de Sales (1877)
1787 Alphonsus Liguori (1871)
1897 Thérèse of Lisieux (1997)
Because of the split between East and West there are no Eastern Doctors after Damascene, making eight Easterners in total. There are three Carmelites (Teresa, John of the Cross, and Therese), two Jesuits (Canisius and Bellarmine), three Dominicans (Thomas, Albert, Catherine), three Franciscans (Anthony, Bonaventure, Lawrence), one Redemptorist (Liguori), and five Benedictines (Isidore [according to tradition], Bede, Anselm, Bernard, Pietro Damiani). There are three women, two of whom were nuns. There are nineteen bishops, of whom two were Popes, two Patriarchs of Alexandria, two Patriarchs of Constantinople, and one Patriarch of Jerusalem. There is one deacon (Ephraem).
It's an interesting game to speculate on who might possibly receive the honor in the future. Here are some possible candidates, in no particular order:
(1) Leander of Seville (d. 600). Pro: The brother of Isidore of Seville (whom he preceded as bishop of Seville) has generally been recognized as a superior writer to his brother, a man of great eloquence who did great work opposing Arianism and united Spain into orthodoxy. He is already allowed the honor in Spain, so it would simply be a matter of extending this universally. Con: Only two works of his have survived, neither well-known. One of them is a monastic rule and the other a homily on the conversion of the Goths.
(2) Fulgentius of Astigi (d. 633). Pro: Isidore's other brother, the middle son in the family, also has the honors in Spain. Con: He is often confused with Fulgentius of Ruspe, and none of his works have actually survived, which seems to be an insuperable obstacle. Serious candidacy would require rediscovery of his writings.
(3) John of Avila (d. 1569). Pro: The Apostle of Andalusia, this student of Domingo de Soto was a profound influence on such luminaries as John of the Cross, Teresa of Avila, Francis Borgia, and Luis of Granada. His works have been widely translated. Con: None, as far as I am aware; Juan de Avila is, I would dare say, the most likely current candidate for the title, or close to it.
(4) Edith Stein (d. 1942). Pro: She's extraordinarily influential, in part on her own merits and in part due to the work of John Paul II, who was heavily influenced by her. Her intelligence and piety were both exemplary. Con: We'll have to see how significant her influence is in a hundred years or so. Also, her death at Auschwitz is regarded for liturgical purposes as a martyrdom, and no martyr has ever received the title. (The reason is fairly simple; 'Doctor Ecclesiae' is a liturgical title. But for liturgical reasons the title of martyr or confessor always trumps the title of doctor, so granted the title to a martyr or confessor would be otiose -- they'd never have the Office of Doctors celebrated for them, because they would always be celebrated as martyrs and confessors.)
(5) Nerses Shnorhali (d. 1172). Pro: His brilliance is undeniable, and his writings have been a profound contribution to the theological heritage of Armenian Catholics. Armenian Catholics, in fact, petitioned John Paul II to declare him a Doctor of the Church. Con: As Catholicos of Armenia he may have worked (unsuccessfully) for reunion with Rome, but he was never strictly Catholic, and is only treated as a canonical saint in the Armenian Catholic calendar.
(6) John Henry Newman (d. 1890) Pro: His influence is clear, and his writings extensive. There is already interest in the idea. Con: He would need to be canonized first, and that can't be guaranteed.
(7) John Paul II (d. 2005) Pro: There's already a push for it. Con: Like Stein, we'll have to see how things stand a century or so from now; and like Newman, he needs to be canonized first (and it needs especially to be reiterated in a case like this that that can't be guaranteed until the final stage of the process).
(8) Jane Chantal (d. 1641) Pro: Jeanne Frances de Chantal was a correspondent of Francis de Sales; she had an extensive correspondence and wrote a number of salutary works on religious life. Con: None, as far as I am aware. I think she is also a highly likely candidate. [It's St. Jeanne's feast day today, by the way.]
(9) Catherine of Genoa (d. 1510) Pro: Her writings have certainly been influential. Con: They are 'private revelations', i.e., descriptions of mystical experiences, and this might not be enough to count at theological doctrine.
No doubt many others could be added.