Correction --- It is still said in the Indult Tridentine Rite (b/k/a the Mass of the 1962 Missal), but not in the Latin Riite (b/k/a the Novus Ordo Rite). Trust me on this one! I grew up as a child in the Latin Rite (now known as the Mass of the 1962 Missal) and for the past 40 years have 'adjusted' to the changes made. In the current Latin Rite, the prayer to St. Michael is no longer used.
Here is the present 'definition' of the Roman Rite liturgies, compliments of EWTN. As a point of information, EWTN's Sunday liturgy is now a high mass, following the Novus Ordo Rite, but totally in Latin. The priests vest in brocade, there are 6 candles on the altar, incense is used in the liturgy. It is extremely dignified and quite beautiful.
• Roman - The overwhelming majority of Latin Catholics and of Catholics in general. Patriarch of this and the other Roman Rites is the Bishop of Rome. The current Roman Rite is that of the 1969 Missale Romanum, to be published in a third edition in 2001.
- Missal of 1962 (Tridentine Mass) - Some institutes within the Roman Rite, such as the Priestly Fraternity of St. Peter, have the faculty to celebrate the sacramental rites according to the forms in use prior to the Second Vatican Council. This faculty can also be obtained by individual priests from their bishop or from the Pontifical Council Ecclesia Dei.
- Anglican Use - Since the 1980s the Holy See has granted some former Anglican and Episcopal clergy converting with their parishes the faculty of celebrating the sacramental rites according to Anglican forms, doctrinally corrected.
• Mozarabic - The Rite of the Iberian peninsula (Spain and Portugal) known from at least the 6th century, but probably with roots to the original evangelization. Beginning in the 11th century it was generally replaced by the Roman Rite, although it has remained the Rite of the Cathedral of the Archdiocese of Toledo, Spain, and six parishes which sought permission to adhere to it. Its celebration today is generally semi-private.
• Ambrosian - The Rite of the Archdiocese of Milan, Italy, thought to be of early origin and probably consolidated, but not originated, by St. Ambrose. Pope Paul VI was from this Roman Rite. It continues to be celebrated in Milan, though not by all parishes.
• Bragan - Rite of the Archdiocese of Braga, the Primatial See of Portugal, it derives from the 12th century or earlier. It continues to be of occasional use.
• Dominican - Rite of the Order of Friars Preacher (OP), founded by St. Dominic in 1215.
• Carmelite - Rite of the Order of Carmel, whose modern foundation was by St. Berthold c.1154.
• Carthusian - Rite of the Carthusian Order founded by St. Bruno in 1084.
Now that it's been some years since I attended the novus ordo, I trust your judgement on this one.
From what I recall though the novus ordo differed markedly depending on whether the priest was tradition-minded or just another raving liberal.
As far as I'm aware omission of the Leonine prayers from the novus ordo was never mandated by the Vatican.
If they are omitted, it's a shame.
The Tridentine Mass and the Novus Ordo Mass are both forms of the Roman/Latin Rite. They are not separate Rites.