Do Catholics really believe in contemporary miracles?
The short answer is yes.
Miracles will never be a “fad” within catholicism. The Church will continue to canonize modern (new) saints and in order to canonize modern saints, the Church will require proofs of a miracle in order to canonize someone to sainthood.
Saint: To be canonized as a saint, ordinarily at least two miracles must have been performed through the intercession of the Blessed after their death, but for beati confessors, i.e., beati who were not declared martyrs, only one miracle is required, ordinarily being additional to that upon which beatification was premised. Very rarely, a Pope may waive the requirement for a second miracle after beatification if he, the Sacred College of Cardinals, and the Congregation for the Causes of Saints all agree that the Blessed lived a life of great merit proven by certain actions. This extraordinary procedure was used in Pope Francis' canonization of Pope John XXIII, who convoked the first part of the Second Vatican Council. - Canonization
The Catholic Church will always believe in miracles, in fact the Church relies on them.
The Catholic Church believes miracles are works of God, either directly, or through the prayers and intercessions of a specific saint or saints. There is usually a specific purpose connected to a miracle, e.g. the conversion of a person or persons to the Catholic faith or the construction of a church desired by God. The Church says that it tries to be very cautious to approve the validity of putative miracles. The Catholic Church says that it maintains particularly stringent requirements in validating the miracle's authenticity. The process is overseen by the Congregation for the Causes of Saints.
The Catholic Church has listed several events as miracles, some of them occurring in modern times. Before a person can be accepted as a saint, they must be posthumously confirmed to have performed two miracles. In the procedure of beatification of Pope John Paul II, who died in 2005, the Vatican announced on 14 January 2011 that Pope Benedict XVI had confirmed that the recovery of Sister Marie Simon-Pierre from Parkinson's disease was a miracle.
Among the more notable miracles approved by the Church are several Eucharistic miracles wherein the sacramental bread and wine are transformed into Christ's flesh and blood, such as the Miracle of Lanciano and cures in Lourdes.
According to 17th century documents, a young Spanish man's leg was miraculously restored to him in 1640 after having been amputated two and a half years earlier.
Another miracle approved by the Church is the Miracle of the Sun, which is said to have occurred near Fátima, Portugal on October 13, 1917. According to legend, between 70,000 and 100,000 people, who were gathered at a cove near Fátima, witnessed the sunlight dim and change colors, and the Sun spin, dance about in the sky, and appear to plummet to earth, radiating great heat in the process. After the ten-minute event, the ground and the people's clothing, which had been drenched by a previous rainstorm, were both dry.
Velankanni (Mary) can be traced to the mid-16th century and is attributed to three miracles: the apparition of Mary and the Christ Child to a slumbering shepherd boy, the curing of a lame buttermilk vendor, and the rescue of Portuguese sailors from a violent sea storm.
In addition to these, the Catholic Church attributes miraculous causes to many otherwise inexplicable phenomena on a case-by-case basis. Only after all other possible explanations have been asserted to be inadequate will the Church assume divine intervention and declare the miracle worthy of veneration by their followers. The Church does not, however, enjoin belief in any extra-Scriptural miracle as an article of faith or as necessary for salvation.
St. Thomas Aquinas, a prominent Doctor of the Church, divided miracles into three types in his Summa contra Gentiles:
Things that are at times divinely accomplished, apart from the generally established order in things, are customarily called miracles; for we admire with some astonishment a certain event when we observe the effect but do not know its cause. And since one and the same cause is at times known to some people and unknown to others, the result is that of several who see an effect at the same time, some are moved to admiring astonishment, while others are not. For instance, the astronomer is not astonished when he sees an eclipse of the sun, for he knows its cause, but the person who is ignorant of this science must be amazed, for he ignores the cause. And so, a certain event is wondrous to one person, but not so to another. So, a thing that has a completely hidden cause is wondrous in an unqualified way, and this the name, miracle, suggests; namely, what is of itself filled with admirable wonder, not simply in relation to one person or another. Now, absolutely speaking, the cause hidden from every man is God. In fact, we proved above that no man in the present state of life can grasp His essence intellectually. Therefore, those things must properly be called miraculous which are done by divine power apart from the order generally followed in things.
Now, there are various degrees and orders of these miracles. Indeed, the highest rank among miracles is held by those events in which something is done by God which nature never could do. For example, that two bodies should be coincident; that the sun reverse its course, or stand still; that the sea open up and offer a way through which people may pass. And even among these an order may be observed. For the greater the things that God does are, and the more they are removed from the capacity of nature, the greater the miracle is. Thus, it is more miraculous for the sun to reverse its course than for the sea to be divided.
Then, the second degree among miracles is held by those events in which God does something which nature can do, but not in this order. It is a work of nature for an animal to live, to see, and to walk; but for it to live after death, to see after becoming blind, to walk after paralysis of the limbs, this nature cannot do—but God at times does such works miraculously. Even among this degree of miracles a gradation is evident, according as what is done is more removed from the capacity of nature.
Now, the third degree of miracles occurs when God does what is usually done by the working of nature, but without the operation of the principles of nature. For example, a person may be cured by divine power from a fever which could be cured naturally, and it may rain independently of the working of the principles of nature. - Miracle