I know that the Tridentine Mass was always said in Latin, but have there ever been exceptions to this rule? I heard that some exceptions were allowed, but have no known sources to back up this statement!
If by “Tridentine Mass” the O.P. means the Mass promulgated by Pope Pius V in 1570 (probably better termed the “traditional Roman Rite”), which was used widely in that form until the liturgical reforms after Vatican II (and is still used to varying degrees today), then the answer is that it is basically only used in Latin, with a small number of exceptions that I am aware of.
Although the Council of Trent did not expressly forbid translating the Mass into the vernacular, it did emphatically assert that the Mass need not be celebrated in the vernacular:
Si quis dixerit … lingua tantum vulgari missam celebrari debere …: anathema sit.
If anyone should say … that the Mass should be celebrated only in the vernacular language …, let him be anathema (Session XXII, Canon 9, my translation).
In practice, however, the liturgical books were only issued in Latin; therefore, this Mass was very seldom celebrated in any language but Latin.
I am aware of one historical exception: in the regions that were part of the old Roman province of Dalmatia (basically modern day Croatia), the Mass was often celebrated in Old Slavonic (which is, however, more or less the Slavic equivalent of Latin).
Mention should also be made of the Missal used in Catholic Anglican ordinariates, published in 2015, which is very close to the traditional Roman Rite. This rite is based on some versions of the Anglo-Catholic liturgy that are, in turn, based on the old Roman Rite.
(Naturally, it is also possible that individuals translated the Mass and celebrated in the vernacular without authorization.)
This order is in traditional-language English ("thee, thou" etc), and it's possible to construct an English-language Mass which is practically identical to the Traditional Latin Mass. We celebrate one such Mass weekly, and have even used it to celebrate a Missa Cantata (traditional ceremonial, but in English). There are some differences as there is quite a bit of the Sarum Use included, and even prayers from the Book of Common Prayer, but aficionados of TLM are satisfied.
The full text is not yet online to link to, unfortunately; and while the printed edition linked to above retails at £65, the full-size version is over £300.
I confess to Almighty God, to Blessed Mary ever-Virgin, to Blessed Michael the Archangel, to Blessed John the Baptist, to the holy Apostles Peter and Paul, to all the Saints, and to you, brethren, that I have sinned exceedingly in thought, word, deed; [he strikes his breast thrice] by my fault, by my own fault, by my own most grievous fault. Wherefore I beg Blessed Mary ever Virgin, Blessed Michael the Archangel, Blessed John the Baptist, the holy Apostles Peter and Paul, all the Angels and Saints, and you, my brethren, to pray for me to the Lord our God.
May almighty God have mercy upon thee, forgive thee thy sins, and bring thee to everlasting life.
Collect for Purity
Almighty God, unto whom all hearts be open, all desires known, and from whom no secrets are hid: Cleanse the thoughts of our hearts by the inspiration of thy Holy Spirit, that we may perfectly love thee, and worthily magnify thy holy Name; through Christ our Lord. Amen.
Prayers at the Offertory (selection)
Receive, O holy Father, almighty and everlasting God, this spotless host, which I, thine unworthy servant, now offer unto thee, my living and true God, for my numberless sins, offences, and negligences; for all here present; as also for the faithful in Christ, both the quick and the dead, that it may avail for their salvation and mine, unto life everlasting. Amen.
Come, thou O sanctifier, almighty, everlasting God, and ✠ bless this sacrifice, made ready for thy holy Name.
Receive, O holy Trinity, this oblation which we offer unto thee in memory of the passion, resurrection, and ascension of our Lord Jesus Christ; in honour of blessed Mary ever-virgin, of blessed John the Baptist, of the holy Apostles Peter and Paul, and all the Saints, that it may be to their honour, and for our salvation; and that like as we remember them on earth, so in heaven they may plead for us. Through the same Christ our Lord. Amen.
Wherefore, O Lord, we thy servants, and thy holy people also, remembering the blessed passion of the same Christ thy Son our Lord, as also his resurrection from the dead, and his glorious ascension into heaven; do offer unto thine excellent majesty of thine own gifts and bounty, the pure victim, the holy victim, the immaculate victim, the holy Bread of eternal life, and the Chalice of everlasting salvation.
Vouchsafe to look upon them with a merciful and pleasant countenance; and to accept them, even as thou didst vouchsafe to accept the gifts of thy servant Abel the righteous, and the sacrifice of our patriarch Abraham; and the holy sacrifice, the immaculate victim, which thy high priest Melchisedech offered unto thee.
Deliver us, O Lord, we beseech thee, from all evils, past, present, and to come; and at the intercession of the blessed and glorious ever Virgin Mary, Mother of God, with thy blessed Apostles Peter and Paul, and with Andrew, and all the Saints, favourably grant peace in our days, that by the help of thine availing mercy we may ever both be free from sin and safe from all distress.
[This answer previously linked to the full-size altar missal and an altar card with some texts, but CTS seem to have withdrawn them.]
Portions of the liturgy were permitted to be celebrated in Native American vernaculars.
If we count the lawful liturgical variants of the Tridentine Mass such as the Dominican Rite and the Carmelite Rite and permission given to some Religious Orders, we able to see that the Holy See has given various indults for the Tridentine Mass to be celebrated in various languages.
Although Latin prevails in the West as a unified liturgical language, in the face of certain circumstances the Roman church has made exceptions to provide a language in the Liturgy more familiar to the people. It is in the ninth century among the Slavic nations that we find a departure from liturgical Latin in divine worship. A privilege was first granted to Sts. Cyril and Methodius, by Pope Hadrian II in 869, and again by Pope John VIII in 880 to use the vernacular (Slavonic) in the Liturgy.
Another example of the flexibility of which the Roman rite is capable is the privilege granted for the use of Chinese as a liturgical language. History records in the fourteenth century that the first Franciscan missionary to China, John of Monte Corvino, used the vernacular in the Liturgy.26 Pope Paul V, in a brief of June 27, 1615, granted the same privilege to Jesuit missionaries. As recently as 1949, the privilege to use the Chinese literary language in the Liturgy was granted by the Holy Office.
Still further concessions have been granted:
a) During the fourteenth century the Roman Liturgy in its Dominican variant was translated into Greek for use by the Dominican missionaries in Greece.
b) Permission had been granted to celebrate the Dominican Liturgy in the Armenian classical language in Armenia.
c) At the end of the sixteenth century missionaries of India of the Latin rite were allowed to celebrate Mass in Syriac.
d) In the seventeenth century the Discalced Carmelites were granted permission to use Arabic in their mission foundation in Persia.
e) In the seventeenth century the Theatine Clerics were granted permission to use Georgian or Armenian in their mission foundation in Georgia.
f) In the nineteenth century the Franciscans in the Holy Land were granted permission to use Arabic.
g) In 1958, an indult was granted India to use Hindi.
h) Five Latin priests in the Holy Land were granted permission to use Hebrew.
i) In 1959, the Holy See renewed Germany's privilege to use the vernacular (German) in the Epistle and Gospel after they are recited in Latin. - Liturgical Languages
Missale Romanum 1965
The official text of the Ordinary is from the Roman Missal with English translations approved by the National Conference of Bishops of the United States, published by authority of the Bishops' Commission on the Liturgical Apostolate.
NOTE: This was the official English version of the Order of Mass from the 1965 Roman Missal, published directly after the Second Vatican Council ended in 1965. This was the English Mass used from 1965 until 1969-70, when Paul VI promulgated the New Order of Mass (Novus Ordo Missae), and imposed it on the Latin Rite (the Novus Ordo is the current normative Mass of the Latin Rite). This interim Mass is much closer to the intended fruit of Vatican II's Sacrosanctum Concilum than the New Mass of 1970. It is essentially the Tridentine Latin Mass in English with minor modifications.
Many rubrical similarities exist between the 1965 Missal and the New Mass of 1970. Obviously, an option for use of the vernacular exists in the1965.Furthermore, as in the Novus Ordo, it is at the discretion of the celebrant to either face the East ("ad orientem") or the people ("versus ad populum"). An option for concelebration was also introduced in the 1965 (this was formerly restricted to Ordination Masses).The required Mass vestments were also simplified (e.g., optionality of the maniple).In 1967, the cope was supressed in the Asperges (rite of aspersion at High Mass). The chasuble was worn in its stead. The Canon was still required to be read in Latin until 1967, when it was permitted in the vernacular. - The Interim Missal