Full Libertarianism is generally accepted as incompatible with Catholic teaching. What parts are generally opposed to Catholicism and the Catechism?
Libertarianism is diametrically opposed to the principle of solidarity.
The principle of solidarity requires that men and women of our day cultivate a greater awareness that they are debtors of the society of which they have become part. They are debtors because of those conditions that make human existence liveable, and because of the indivisible and indispensable legacy constituted by culture, scientific and technical knowledge, material and immaterial goods and by all that the human condition has produced. A similar debt must be recognized in the various forms of social interaction, so that humanity's journey will not be interrupted but remain open to present and future generations, all of them called together to share the same gift in solidarity.
The straw-man libertarian would say that "a man who pulls himself up by his own bootstraps" doesn't owe anybody anything. But Rome acknowledges that very critical fact that she wasn't built in a day and says we all owe each other some sort of debt. Which is probably why Jesus said "forgive us our debts" instead of "forgive us our trespasses, failings, etc..". We can understand "forgiving other peoples trespasses against us", and that is good, but on a super deep level, what we're saying is an non-ironic "excuse me for living"
Libertarianism is not diametrically opposed to the principle of subsidiarity. They get it right there, with respect to Federal rights, State rights, individual rights. Individual rights are at the top. Subsidiarity is the principle that government is done best at the appropriate level. Libertarianism argues that families should be free from government interference, that is in line with Catholicism.
But it might tend to go a step farther and stress individual rights over the Catholic notion that the "Family is the fundamental unit of society". Catholicism acknowledges only the usual family unit borne out of marriage with openness to children. Libertarianism doesn't necessarily favor one form of a family over another. Catholicism says that only a family modeled on the Holy Family will prosper and other forms of a family units are to be avoided.
One of the challenges of libertarianism is that human nature is depraved in many ways.
Without some sort of state incentive to help the poor & vulnerable, or state corrective against predatory practices that take advantage of the poor & vulnerable, the principle of subsidiarity (also shared by many non Catholics) is at risk.
Personally I think that the best way to help those stuck in poverty is to encourage a free market emphasis on education & entrepreneurship. Toxic charity & paternalistic models of humanitarian mission work is something to be avoided. As much as possible, hand outs should be linked with being able to give hand ups.