Does anyone know how often the popes have spoken infallible?
The actual pronouncement by Blessed Pius IX of infallibility itself.
The doctrine of the Immaculate Conception, again by Blessed Pius IX.
The doctrine of the Assumption, by Pius XII.
I believe that canonizations are also regarded as infallible.
There was a thread herein that went on at length about how JP2 spoke infallibly when he issued forth on the inablity of the Church to ordain women. However, there was much discussion within the theological community as to whether or not that was an ex-cathedra statement, and the consensus was that it was not (amatuer theologians lurking in these threads to the contrary).
The only two accepted times a Pope has spoken ex-cathdra has been about the Immacualte Conception and the Assumption, to the best of my knowledge. And on that, I rely on better trained theologians than the lurkers.
As a sub-set of this point, there has been, and continues to be, an ongoing discussion (if you will) concerning the whole issue of authority of the Pope and the bishops, and coupled with that, a discussion of ecclesiology; that is, exactly what is the structure of the Church. There are those who want a very rigid, hierarchical model with the Pope as head, and the bishops reduced to something close to rubber stamp minions. There are others, on the other end of the spectrum, who would reduce the Pope to a figurehead, nice to trot out when there is a special occasion, and pretty sharp looking iin those white robes, but incapable of leading at all, let alone effectively. Vatican 1 dealt more with the Pope; and Vatican 2 dealt more with the bishops. Neither really had the last word to be said on the issues.
Another issue is that most people are not too aware of how the Pope relates to the universal Church (which they usually take to mean the Roman rite). The Eastern rites too often are presumed to be nothing more than people from “over there” who use a lot of incense and have differeent vestments, and us foreign languages. the reality is a bit more complex than that. And while that does not have to do directly with infallibility, it does have to do with when and how the Pope may invoke infallibility.
Note: the above has to do with the Pope exercising the charism of infallibility in an ex-cathedra statement, and not as to his exercise of infallibilty as the head teacher in terms of the Magesterium, or through canonization.
Dogmatic statements from Church councils that have been ratified by the Pope also form part of the extraordinary magesterim, and thus are infallible. Such decrees usually end with an anethema: “Whoever says such and such, let him be anathema”, etc. When a Council issues such a decree, and when the decree has been ratified by the Pope, it becomes an infallible statement of the magesterium. We have hundreds of such dogmatic statements.
There have been several times (at least one that I know of) when a council gave a dogmatic pronouncement that ended with a “let him be anathem”, which was not ratified by the Pope.
The second Vatican Council did not give any such statements, but it did quote some previously defined dogmatic statements. No new doctrines were defined at Vatican II. It was the first general council of the church that was merely pastoral in nature (no infallible dogmas issued), rather than dogmatic.
There are several documents of Vatican II that have the name “dogmatic” in them - “Dogmatic constitution of the Church”, and “Dogmatic constitution on Divine Revelation” - but no dogmatic statements were given at Vatican II.
Just after the close of the Council, Paul VI himself confirmed that Vatican II was not an infallible council when he said: “Given the Council’s pastoral character, it avoided pronouncing in an extraordinary manner, dogmas endowed with the note of infallibility”(Paul VI, General Audience of January 12, 1966).