According to the rubrics, priests are supposed to wear their biretta during certain portions of the Mass, such as when they enter or leave the sanctuary, when they are sitting off to the side, and when preaching the sermon. They do not wear the biretta when standing or kneeling (with the exception of preaching the gospel), it is also removed when the Holy Name is mentioned -- according to Fortescue and O'Connell, The Ceremonies of the Roman Rite Described, pp. 52-53, 279
I believe the exception for preaching is because the sermon is not properly part of the Mass. Technically, the priest is supposed to remove not just the Maniple but also the Chasuble to preach the sermon (then put on his Biretta) although I have only seen this level of liturgical precision at St. Gertrude the Great Church in West Chester, and among indult orders.
The S.S.P.X, however, follow the local custom in certain parts of France where the Biretta is not used at all. This is the reason for their lack of birettas as it has been explained to me. Many priests in North America seem to use the Biretta just for entering and leaving the sanctuary, although at least in Canada I have seen the Birettas disappear entirely whenever someone from Europe (such as Fr. Pfluger or Bp. Fellay) is in town.