If I were a Catholic, I would probably be a Sedevacantist

"Sedevacantism" is a fringe movement within Catholicism, held by some Catholics, including some priests, that posits that all the recent popes are heretics and, thus, are not genuine God-appointed heads of the Catholic Church and, thus, the position of the Pope, ie. the Holy See, has been vacant for several decades, with no genuine God-approved Pope holding the office.

Which Pope was the last legitimate one depends a bit on the sedevacantist, but the most general consensus is that it was Pius XII (who was Pope from 1939 to 1958), and that the first non-legitimate heretical pope was the next one, John XXIII, under whose papacy the famous (or rather, infamous, from the perspective of sedevacantists) Second Vatican Council was held.

(There's probably no consensus on whether John XXIII started as a legitimate Pope and later became a heretic, or whether he was a heretic from the start, but that doesn't really make a lot of difference. Even if he was legitimate at first, in the sedevacantist view he lost his claim to the Holy See when he committed heresy, and thus stopped being the legitimate Pope.)

Sedevacantists do not deny the teachings of the Catholic Church nor the authority of the Pope (as long as he is the legitimate one). On the contrary, they tend to be extreme hard-line Catholics who very strongly believe in traditional Catholicism, and that the Catholic Church, with all of its teachings, has been the True Church of Christianity for two thousand years.

However, they deny the validity of many of the decrees of the Second Vatican Council, considering most of it heretical, and with it likewise all the subsequent popes who have accepted those decrees to likewise be heretics and thus not true Popes.

The Second Vatican Council was one of the most notorious ecumenical councils of the Catholic Church, generating fundamental sweeping changes to many of the official positions and teachings of the Church. In many ways this council made the Catholic Church significantly more "liberal" in its views, away from the very strict hard-line positions held previously. Some of the most significant changes were:

• Catholic Mass had been held in Ecclesiastic Latin for over a thousand years. This council freed all priests to allow them to hold Mass in their local language.
• The position that the Catholic Church is the only true Church of God was softened and replaced with a more ecumenical view where Catholicism acknowledges that other denominations may also hold some validity in the eyes of God.
• The council declared that not all Jews were responsible for the execution of Jesus.
• It also declared that all people have religious freedom, the fundamental right to choose which religion they follow (in other words, the Catholic Church cannot force nor coerce people into following Catholicism.)
• The position on studying the Bible by laypeople was changed: While previously the Church discouraged laypeople from studying the Bible without guidance and supervision from a priest, now the position was reversed and, on the contrary, laypeople were encouraged to do read the Bible even if there is no priest present.
• And related to that, the approach to scriptural interpretation was changed, encouraging taking into account the historical and cultural context of when the scriptures were written, when interpreting their meaning. (In other words, a more or less subtle hint that "not everything written should be taken literally, as if it applied in the exact same way to the modern world. Context matters.")
• The council somewhat de-emphasized the supreme authority of the Pope in all matters concerning the Church, and increased the role of bishops in those matters. (In other words, bishops became more free to make decisions without necessarily having to seek approval from the Vatican for every little thing.)
• Several changes were made to liturgies and church paraphernalia. 

At face value most of those changes seem positive. And, in fact, the Second Vatican Council was organized precisely because the pope of the time, John XXIII, felt that the Church was in need of such reforms. He felt that the animosity between the Catholic Church and other denominations had to end, as well as many other sweeping changes to make the Church more "people-friendly".

Sedevacantists, however, argue that most of those changes, and the entire motivation behind the council, were motivated by secular ideology, secular politics, and secular movements, rather than by God. They argue that John XXIII was not inspired by God to initiate this kind of radical reform, but by secular politics (and, thus, ultimately, Satan himself.)

After all, what is more likely, that the Catholic Church was completely wrong on those issues for almost two thousand years and only now, after all this time, and coincidentally alongside the rest of the world, decided to reveal the "actual truth" about those matters, or is it more likely that the Pope was inspired by secular politics rather than God?

Why would God wait for almost two thousand years, allowing the Church to have the wrong teachings and positions, before revealing the actual truth? And what a coincidence that this actual truth was revealed at the same time that the rest of the world was moving towards more liberal politics. Did God just come to the Pope and tell him "hey, you know what? Yeah, those secular philosophers, politicians and activists are actually right. Oops, my bad. I suppose we were both wrong. You'd better reform the Church."

Sedevacantists consider several of those changes to the official teaching and positions of the Church to be heretical, as they contradict the official positions held by the Church for almost two thousand years. And, consequently, they consider any pope who accepts those heretical views to be a heretic himself and, thus, not an actual legit valid Pope. Since none of the subsequent popes has denounced those reforms, they have all been heretics, and thus the Holy See has been vacant since that council.