Which Catholic doctrines are infallible?

Jayarathina Madharasan has already answered the question asked. But I noticed there is another sub question to this which is: "I can't immediately think of a reason why ex cathedra statements would be the only ones to result in infallible doctrines"

I would try to put some light on this valid point.

Ex-cathedra means "solemn papal definitions". It is the Charism the Pope "enjoys in virtue of his office" or literally it means "from the chair".

Many confuse charism of Infallibility as impeccability. Some imagine that Pope cannot sin. Infallibility is not the absence of sin nor is it a charism that belongs only to Pope. Indeed, infallibility also belongs to the body of bishops as a whole, when, in doctrinal unity with Pope, they solemnly teach the doctrine as true.

On infallibility, Vatican II explains that

"Although the individual bishops do not enjoy the prerogative of infallibility, they can nevertheless proclaim Christ’s doctrine infallibly. This is so, even when they are dispersed around the world, provided that while maintaining the bond of unity among themselves and with Peter’s successor and while teaching authentically on a matter of faith or morals, they concur in a single viewpoint as the one that must be held conclusively. This authority is even more clearly verified when, gathered together in an ecumenical council, they are teachers and judges of faith and morals for the universal church. Their definition must then be adhered to with submission of faith (Lumen Gentium 35)."

Some argue that how popes can be infallible if some of them lived scandalously. That is why it is infallibility not impeccability. The doctrine don’t say that Pope wont’ sin or give bad example. Some wonder hoe infallibility could exist if some popes disagreed with others. In fact the accurate understanding of infallibility is that which applies only to solemn, official teachings on faith and morals, not on disciplinary decisions or even to unofficial comment on faith and morals. A Popes private theological opinion are not infallible but only that which is solemnly defined is considered infallible teaching.

Some think that infallibility means that popes are given some special grace that allows them to teach positively whatever truth needs to be known. That is not correct either. Infallibility is not a substitute for theological study on the part of Pope.

What infallibility does do is to prevent Pope from solemnly and formally teaching as truth something that is in fact an error.

The idea that everything a Pope (or the Catholic church) teaches is infallible is simply a flawed premise.

After searching on the Vatican website, I found that the CCC quite clearly defines dogma, which is the form of doctrine which is infallible or irrevocable in the words of the CCC (emphasis mine):

The dogmas of the faith

88 The Church's Magisterium exercises the authority it holds from Christ to the fullest extent when it defines dogmas, that is, when it proposes, in a form obliging the Christian people to an irrevocable adherence of faith, truths contained in divine Revelation or also when it proposes, in a definitive way, truths having a necessary connection with these.

89 There is an organic connection between our spiritual life and the dogmas. Dogmas are lights along the path of faith; they illuminate it and make it secure. Conversely, if our life is upright, our intellect and heart will be open to welcome the light shed by the dogmas of faith.

90 The mutual connections between dogmas, and their coherence, can be found in the whole of the Revelation of the mystery of Christ. "In Catholic doctrine there exists an order or hierarchy of truths, since they vary in their relation to the foundation of the Christian faith."

But the Catholic teaching on and use of ex cathera dogmatic declarations is difficult to unravel. In the history of the Catholic Church I can find authoritative references (wikipedia is not authoritative) for only three doctrines which are undisputed to have been declared ex cathedra dogma:

• The Immaculate Conception of Mary (declared by Pope Pius IX in 1854, CCC 491 & 966)
• The Bodily Assumption of Mary into Heaven (declared by Pope Pius XII in 1950, CCC 966)
• The Holy Trinity (CCC 249 - 256)

That appears to be it. Searching the Vatican website we can turn up other references to possibilities:

• Dogma of Papal Infallibility (referenced by many papers on the Vatican.va site)
• Dogma of original sin (but specifically mentioned as disputed)
• Dogma of the Consubstantiality (proposed by Pius XI, uncertain if ratified by the bishops)

Though I can't find a reference, I would expect that all creedal statements of church councils which are recognized as valid by the Catholic church are considered dogmatic.

To further muddy the water, many church doctrines are widely considered dogmatic while never having been officially declared as such; these are popularly treated as dogmatic and have an extremely low possibility of ever being rescinded (and the creeds may be in this category).

The main point is that truly ex cathedra dogmatic and irrevocable doctrines are few and far between.

All other doctrinal positions are subject to modification and may be rescinded, even the "ordinary" conclusions and declarations of church councils. That is not to say they are to be taken lightly; papal and official statements from the church are to be considered binding on all Catholics for as long as they are in force -- but they are not automatically infallible.

See this CSE answer and this article from uscatholic.org and this Vatican statement on infallibility for more details. The last of these explicitly mentions the only two undisputed ex cathedra statements made:

As you know there are cases in which the papal Magisterium is exercised solemnly regarding particular points of doctrine belonging to the deposit of revelation or closely connected with it. This is the case with ex cathedra definitions, such as those of Mary's Immaculate Conception, made by Pius IX in 1854, and of her Assumption into heaven, made by Pius XII in 1950. As we know, these definitions have provided all Catholics with certainty in affirming these truths and in excluding all doubt in the matter.

Specifically note the words: "... there are cases in which ...".

From what I have been able to ascertain, it seems like quite specific verbiage introduces an ex cathedra declaration, along the lines of:

... by the authority of Jesus Christ our Lord, of the Blessed Apostles Peter and Paul, and by our own: We declare, pronounce, and define that ...

Which Catholic doctrines are infallible?

Catholic Doctrines are not necessarily infallible. Only defined dogmas are infallible.

Certain doctrines, such as the doctrine of the Limbo of Infants, are not infallible. Papal encyclicals are likewise not necessarily infallible.

Encyclical

A papal document treating of matters related to the general welfare of the Church, sent by the Pope to the bishops. Used especially in modern times to express the mind of the Pope to the people. Although of themselves not infallible documents, encyclicals may (and generally do) contain pronouncements on faith and morals that are de facto infallible because they express the ordinary teaching of the Church. In any case, the faithful are to give the papal encyclicals their interior assent and external respect as statements of the Vicar of Christ.

An encyclical epistle is like an encyclical letter but addressed to part of the Church, that is, to the bishops and faithful of a particular area. Its contents may be doctrinal, moral, or disciplinary matters of universal significance, but may also commemorate some historical event or treat of conditions in a certain country or locality.

Dogmas on the other hand must be believed.

Dogma

Doctrine taught by the Church to be believed by all the faithful as part of divine revelation. All dogmas, therefore, are formally revealed truths and promulgated as such by the Church. they are revealed either in Scripture or tradition, either explicitly (as the Incarnation) or implicitly (as the Assumption). Moreover, their acceptance by the faithful must be proposed as necessary for salvation. They may be taught by the Church in a solemn manner, as with the definition of the Immaculate Conception, or in an ordinary way, as with the constant teaching on the malice of taking innocent human life.

There are too many Catholic dogmas to list here and many have already been posted in other answers. Catholic dogmas may be pronounced ex cathredra by a reigning pope or defined by a Church Ecumenical Coucil.

You might want to look at the book "Fundamentals of Catholic Dogma" by Ludwig Ott. In addition to saying what the Catholic Church teaches, it annotates the teachings with indications of how certain they are and (therefore) how obligatory it is to believe them. The highest level is "De fide" ("of the faith"); these are doctrines infallibly proclaimed by popes or ecumenical councils as part of the deposit of faith and therefore to be believed "with Divine and Catholic faith". (If you deny one of these doctrines, you're a heretic.) The introductory part of the book also contains an explanation of the other levels of certainty, such as "sententia fidei proxima", "sententia certa", and so forth.