In the liturgical context, I think the Liturgiam Authenticam, which provided guidance on how to conduct translations, is pretty clear on this:
[I]n accordance with immemorial tradition, which indeed is already evident in the above-mentioned “Septuagint” version, the name of almighty God expressed by the Hebrew tetragrammaton (YHWH) and rendered in Latin by the word Dominus, is to be rendered into any given vernacular by a word equivalent in meaning.
In English, the tetragrammaton should be replaced with "LORD", although I've seen "God" used as well.
However, outside of that context, the Church is less strict. In the follow-up letter, the Congregation for the Doctrine of the Faith made a strong case that the use of the tetragrammatron is philologically misguided:
The Greek translation of the Old Testament, the so-called Septuagint, dating back to the last centuries prior to the Christian era, had regularly rendered the Hebrew tetragrammaton with the Greek word Kyrios, which means "Lord." Since the text of the Septuagint constituted the Bible of the first generation of Greek-speaking Christians, in which language all the books of the New Testament were also written, these Christians, too, from the beginning never pronounced the divine tetragrammaton. Something similar happened likewise for Latin-speaking Christians [...] in these translations, too, the tetragrammaton was regularly replaced with the Latin word "Dominus", corresponding both to the Hebrew Adonai and to the Greek Kyrios. [...]
This fact has had important implications for the New Testament Christology itself. When in fact St. Paul, with regard to the Crucifixion, writes that "God has highly exalted him and bestowed on him the name which is above every name" (Phil 2:9), he does not mean any other name than "Lord" [...]
but it issued no guidance about whether it should be used elsewhere, stating only (emphasis mine):
Avoiding pronouncing the tetragrammaton of the name of God on the part of the Church has therefore its own grounds. Apart from a motive of a purely philological order, there is also that of remaining faithful to the Church's tradition, from the beginning, that the sacred tetragrammaton was never pronounced in the Christian context nor translated into any of the languages into which the Bible was translated.
Outside of a Christian context, it may be like nails on the chalkboard for someone who perhaps gets annoyed when people talk about the Brontosaurus, but there is no direct prohibition about using it. Of course, it would be a sign of respect to hum it or replace it the word Lord if at all possible.