In which languages could the Tridentine Mass be said? 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.)