[A]s G.K Chesterton said, "If you hear a thing being accused of being too tall and too short, too red and too green, too bad in one way and too bad also in the opposite way, then you may be sure that it is very good."
Therefore, I urge all those interested in defending the Faith to jot down every conceivable argument you have ever heard against the Faith. For as you do you will find a fun and interesting thing happening: virtually every one of these arguments has a polar opposite which is also used to berate the Church.
[Here's a] fun set of twin complaints against the Church. The medieval Church, I was informed by a recent movie review, was a sinkhole of superstition and cowering obeisance to ignorant belief in the supernatural. It was not until the Renaissance and the Scientific Revolution brought the light of Rational Truth to the world that the Church's hoodoo of priestcraft and supernaturalism was broken.
Good Enlightenment dogma that. Yet imagine my surprise when, a week later, the same film critic (this time reviewing a movie about Nostradamus) informed us all that Nostradamus was a mystic in touch with the rhythms of the supernatural and open to the larger spiritual realities which pervade our existence. Sadly, however, he was rejected by a rationalistic, linear, logic-chopping Western Church that was totally closed to the reality of the supernatural.
In short, any stigma will do to beat a dogma. However, once one gets the hang of it, one can have a jolly time cataloging the complaints leveled against the Church and matching them up like bad blind dates with contradictory anti-Catholic complaints. Then, when they have obliterated one another like matter and anti-matter, a Catholic apologist can step into the eerie silence created by the fuddled confusion and state the actual Catholic position.
Here are some more standard canards. Consider them a "starter kit" for your collection. Be the first Catholic on your block to get the whole set!
Lots of great examples follow.