I found this article in Texas Monthly, about Holy Family American Catholic Church here in Austin, to be particularly interesting. The 'American Catholic Church in the United States' is an example of what is known as 'Independent Catholicism' -- High Church Protestantism, in fact, although they sometimes get offended when you point out that there is literally no difference between them and the Episcopalians except that the latter are better at it. There are lots of little splinter groups of this sort; I hadn't heard of this particular one before, but it is of the usual pattern. These religious movements survive by a process of sweeping up people alienated -- for any of infinite number of reasons -- from their Catholic communities and promising a more congenial atmosphere. One can always predict offhand how they will describe it -- more compassionate, more inclusive, more relevant to the modern world. Not all do, but those that officially allow contraception or celebrate same-sex marriage or ordination of women advertise it. And the predictability is not surprising; they are in fact the liberal reflections of their conservative opposites, sedevacantists (which, contrary to some classifications, I do not consider Independent Catholics, for a number of reasons too complicated to get into), and exhibit much the same range -- and lack of range -- and for the same reason that if they weren't within that limited range of options, they would be in communion with Rome or not be calling themselves Catholic. There are only so many things you can be if you insist on being neither hot nor cold.
The world of Independent Catholicism or Breakaway Catholicism or Pseudo-Catholicism -- as a Catholic would certainly consider them -- is a very complicated one, and there is no general formula for evaluating them. The most massive group are churches linked by the Bonn Agreement, which guaranteed sacramental intercommunion between the Anglican Communion and the Union of Utrecht (Old Catholics, as they are sometimes called), although sometimes these are not given the actual label of Independent Catholic. The Union of Scranton (consisting primarily of the Polish National Catholic Church and the Nordic Catholic Church), which is not part of the Bonn Agreement, is somewhat more conservative; the PNCC originally broke away due specifically to a real failure of American bishops to provide adequately for the needs of Polish immigrants, so it has drifted far less than most Independent Catholic churches. (This is a general pattern; Breakaways arising from specifically identifiable injustices, perceived or real, tend to drift very slowly around where they started, while Breakaways of a more general type tend to accelerate away.) All of the PNCC's sacraments, while illicit, are consistently valid, which is no longer true of the Union of Utrecht. PNCC is a Canon 844 §2 church, which means that Roman Catholics may sometimes receive Eucharist, Reconciliation, or Unction from the PNCC in emergency situations, whereas Union of Utrecht churches are not -- individual ministers may have legitimate orders, and thus valid sacraments, but no general guarantee of this exists. The ancient Apostolic Churches are all 844 §2, while Breakaways are very rarely so, and thus the distinction ends up being a quite significant one.
The Ecumenical Catholic Communion, farther out still, is probably the largest coherent mass that is not part of one of these communions.
Outside these, though, the label is a grab-bag of many different splinters. The American National Catholic Church and the American Catholic Church in the United States -- which are not the same -- are each big on particular liberal interpretations of the Second Vatican Council; the Antiochian Catholic Church in America has a mix-and-match of Oriental Orthodox practice and theology. The Iglesia Católica Apostólica Mexicana, which is the one that actually annoys me, is a church invented by the government of Mexico in 1925 amidst the persecutions of Catholics that led to the Cristero War. If there is any Independent Catholic denomination whose existence defies all reasonableness and decency, it is the quisling ICAM.
It's an interesting phenomenon. It's a very old one as well. Most Breakaways through the centuries have tended to fade away unless they have secular support, but they have always existed, and inevitably arise when catechesis or priestly formation or episcopal teaching are bad, or when secular powers decide that having their own particular church would be easier than dealing with a universal Church. The problem they always face is that there doesn't seem to be a path that's neither Protestant nor parasite -- that is, they all tend either to become indistinguishable from Protestants or they survive only by continually picking off alienated Catholics. I suspect that we will see more of them in the near future; build-your-own-church is a very powerful temptation.