The real answer to the questions is that aren't supposed to be any factions. The Church, after all is Catholic, universal. Few things are more damaging to the Church than dividing into factions based on one pet cause or another.
Officially the most obvious divide is that there are 22 Rites withing the Catholic Church, of which about 98% of Catholics belong to the Roman Rite. This is mostly on a geographic basis and most of the other 21 rites are in remote areas of Eastern Europe. Immigrants from there will tend to stay in their rite if possible, but many simply assimilate into Roman Catholicism if their rite doesn't exist where they live. In places with large numbers of Eastern European immigrants, such as the Cleveland area, there are a dozen or so rites that have parishes.
Apart from that, Parishes were traditionally defined by what neighborhood you lived in. In the US, this has historically resulting in divisions by ethnic group. For instance, I was raised in the "German Catholic" church in my hometown. Even today when it is clustered with two other parishes and shares the same priest, the majority of people who go to the "German" church are of German ancestry and the people who go to the Irish or Italian churches are mostly of Irish or Italian ancestry. That was how the divisions were set up for most of American history.
Now today the ethnic lines are becoming less important and people move around a lot more. In Madison, where ethnic background is irrelevant, people will mostly go to the parish nearest them. There are occasionally stylistic differences from parish to parish, some will have their act together better than others, and have slightly different styles. There is normally a wide range of style between the Masses at a given parish too.
Of course, those aren't the types of factions people reading political forums are generally thinking about
. First there are the Catholics who no longer believe in the Church's teachings on x, and will sometimes identify in such a way. We do get the occasional radical traditionalist who will identify as a "traditionalist Catholic" or something along those lines, but generally only if you know where to look for them. Some of them are schismatic groups that aren't Catholic, so there's that too. You may find a similar phenomenon will "charasmatic Catholics". However, those sorts of groups are not supposed to be factions, but rather just Catholics who think they should go to a Tridentine Mass or a charasmatic Mass. If those truly do become factions, then they're doing it wrong. The Catholic Church is not supposed to have factions, that's part of the point: putting away all our little special interests to be a member of the Church founded by Christ, not the church founded by TJ (etc.).
In Catholic grad student-land, the real argument is generally between the people who think we ought to work with American conservatism and those who believe we can work with no one. It's an argument between the Scalias of the world and the Third/Way/Other types. We aren't neatly fit into that paradigm and most of us are somewhere in between. I'm certainly in the middle of that spectrum.
