What is an overview of Christian doctrines on the extent to which the Law of Moses still applies?

***Note**: this question is inspired by the question https://christianity.stackexchange.com/q/456/50422 . Since that question was closed as off-topic, I'm asking this new one in an attempt to contribute with an on-topic version of the former.* The Law of Moses is a collection of about [613 commandments](https://en.wikipedia.org/wiki/613_commandments) , according to the Jewish tradition. The question about the extent to which those commandments still apply today (and to what groups of people) is a very important one, as well as the source of multiple doctrinal disputes and denominational divisions among Christians. For example, some believe that the entire Mosaic Law is still morally binding, universally, on all people. Some believe that the Mosaic Law only applies entirely to the Jews, whereas the Gentiles are exempt from this obligation and only required to keep a subset (and which subset?). Some believe that only the 10 commandments are morally binding, with the exception of the Sabbath. Others believe that the 10 commandments are biding, including the Sabbath. Others go one step further and keep both the Sabbath and the feast days. There are also disputes about the contemporary applicability of tithing, the dietary laws, etc. What is an overview of Christian doctrines on the extent to which the Law of Moses still applies? How do different denominations decide which commandments are still applicable and which aren't?