Question: Sometimes there are prayers recited after Mass (three Hail Marys, Hail Holy Queen, etc.) and sometimes not. Is there a rule governing their recitation?
Answer: The prayers in question are called the "Leonine" prayers, after Pope Leo XIII who first ordered in 1884 that they be said "for the needs of the Church." Popes Saint Pius X and Benedict XV renewed Pope Leo's prescription in 1903 and 1915. On June 30th 1930, Pope Pius XI ordered that they be said for the conversion of Russia.
We paraphrase the instructions given by Fr. J.B. O'Connell, who cites numerous decisions of the Sacred Congregation of Rites as to when the prayers may be omitted:
They are to be recited after a "private" Mass, so they never follow solemn or sung Mass. Additionally, they may be omitted:
1. After a low Mass which takes the place of a Solemn Mass (e.g. a low Mass of ordination, episcopal consecration, burial, the community Mass of a monastery or a cathedral chapter). After a Mass having the privileges of a solemn votive Mass (e.g. of the Sacred Heart on First Fridays, or Christ the Eternal High Priest on the First Thursday or Saturday of the month).
2. After certain low Masses offered with "additional solemnity" (e.g. a wedding, Confirmation, general Communion, profession of vows, first Mass of a newly ordained priest, a jubilee), and on Candlemas, Ash Wednesday, Holy Thursday, and Holy Saturday.
3. If a "sacred function or pious exercise" immediately follows Mass (e.g. Benediction, Holy Communion, a sermon, distribution of ashes, etc.).
4. If two or three Masses are said without interruption on All Souls Day, the leonine prayers are said only after the last Mass [assuming that it is not sung]. Likewise the second and third Masses of Christmas if offered without interruption.
5. If the bishop prescribes additional prayers, they follow the Leonine
1. J.B. O'Connell, The Celebration of Mass: A Study of the Rubrics of the Roman Missal (Milwaukee: Bruce, 1959), pp. 172 - 174.