It is completely falling apart as modern science continues to provide facts that disprove believe. The PR blow that Catholicism has taken in the past decade, mixed with a surprising study done earlier this year concluding that atheists are on the rise leads me to believe the Catholic Church is seeing it's final days. When your organization argued that they receive their information from God, yet they thought the earth was flat at the time, that should give you an indication as to whether or not they were truly speaking with a deity.
Catholicism, like most religions, is faltering because people are turning away from it. Rates of non-religious people are rising throughout the world, with non-religious making up 1/3 of the world's population. Catholicism is also unstable in that it often-times contradicts science and logic in its dogma, and it has been very slow to accept this.
Catholicism and the Catholic Church will weather the current storms of secularism, hedonism and atheism. It has many times before. Catholicism is currently in the shadow, hidden by the monstrosity emanating from Vatican City. All of it will pass. The Catholic Church will survive, perhaps not as a large percentage of people, but it will always be stable.
I am not even sure what this question means. Are we asking if the church itself is unstable? If its beliefs are in question? Or if those who follow Catholicism unstable? Any way you look at it, the answer is no. Yes, times are changing. They always change. The Catholic church is actually one of the few churches that changes with the times. Are there problems? Yes, of course. That is the nature of belief. But is it going to collapse? I don't think so.
The Catholic church is as stable as it ever was. Although there has been some decline in attendance, The Church is still vibrant and relevant in today's society. The hierarchy is still in place, and works well in most cases. Catholics know who to go to with various issues, and most feel comfortable with things the way they are.