There is nothing new under the sun. The same heresies that were condemned by the early Church are alive and well today (Arianism, Gnosticism, Nestorianism, Pelagianism etc., etc.)
In modern terms I would say the following are what jump out at me in this age:
- Once saved, always saved.
- Eucharist is only a symbol.
- Denial of the Trinity
- Baptism as a public proclamation of faith rather than a cleansing of the soul.
- Denial that the Church has the power to forgive sins.
- Denial of the sacraments in general.
- Denial of the authority given to the Church by Christ (power to bind and loose).
I am sure, seeing as there are at least 30,000 Christian denominations, that there are many, many more, some of which will come to mind after I post this.
With the exception of the number three, which incidentally I have never encountered in the Catholic Church, this list is pretty Protestant in nature.
The errors (I’ll refrain from calling them heresies, though likely they sometimes reach that point) that I’ve noticed among Catholics in my neck of the woods include:
-Support for contraception, self-abuse, homosexuality, and other such sexual sins
-Support for some forms of mercy killing
-Complete pacifism
-Warped interpretation of Catholic social teaching that essentially reduces it to either the Democratic or Republican party platform
-Various forms of indifferentism regarding world religions or at least other forms of Christianity
-Extreme, emotionally charged rejection of wholesome things like scholastic philosophy, older Western liturgical forms, certain saints, etc.
-Experimentation with South or East Asian forms of meditation or pseudo-science
-Emphasis on the humanity of Christ to the point of harming recognition of His divinity.
-A paradoxical combination of moral relativism and moral absolutism, depending on which approach suits their passions.
-No doubt arising from this last error, a vague rejection or minimization of Papal authority, and of Church authority in general, combined with an almost fanatical zeal for obedience to the Church on issues they personally agree with and feel strongly about.