Dochawk definitely gives the biggest example of how the dogma is misunderstood.
Another example - that actually flows from Doc’s example - is the misguided belief that any time the pope speaks or sets pen to paper theological gold pours forth in abundance.
We were blessed in that our two previous popes were both brilliant theologians. That sort of set a standard in the minds of many Catholics. Now that our current pope isn’t quite the academic that our previous two were, it has many people confused. Don’t get me wrong, Pope Francis has his own “genius,” and I respect him for that. But his is a very different “genius” from St. John Paul II and Pope Benedict XVI.
Another misunderstanding is that every document written by the pope must necessarily be infallible. It is forgotten that papal documents themselves have their own hierarchy and not all are of equal weight.
And finally there is this mistaken notion that the pope can just wake up one morning and make an ex cathedra statement without any sort of context, and without consulting his brother bishops. It’s forgotten that the pope works with the bishops as much as the bishops work with him, and ex cathedra statements are a response to an ongoing theological controversy that has reached a boiling point.