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

I think the question is asking for a list of different opinions about what Christian groups think which aspects of the Mosaic Law are in force, meaning the original obligation and covenant is unchanged and therefore the original requirements are binding as they were in the original words communicated by God to Moses.

The Bible does not actually distinguish within the Mosaic law a real division of moral, civil and ceremonial, as though one part could be no longer binding while another part could still be. Either a person is under the whole law or not under any part of the Law. Either the whole Law has been fulfilled, or none of it. However, it is convenient to look at it that way for the purpose of answering the question because the question is about group who think they can do that, and what are those groups and ways that they do that.

For the ‘civil law’, people usually mean the ‘national governing aspect’ over Israel and more specifically in practical terms just the ability of the law to execute those who break it by stoning, strangling or other forms of civil control. This civil power is to prevent taking of ‘the law’ into ones own hands, for example, in revenge. Of course, in no denomination is this practiced today.

Second, for the ‘ceremonial’, it refers primarily to the sacrifices of animals in the temple according to the Aaronic Priesthood. Well again nobody practices the ceremonial law today otherwise it would be difficult to find lamb at the local restaurant.

Finally we come to the topic at hand - the moral law. Here the church is split in about three high-level ways.

The three splits are about how, after believing in Christ, does a person satisfy all the just requirements of the Law and avoid its deadly condemnation. The split starts here, not around which specific law is binding but how as a whole, how one can satisfy its unforgiving demands.

First there is the reformed view as proposed by people like Martin Luther, who see the entire Law satisfied by Christ’s death, so that the entire law is fulfilled in a single act, a personal moment in time, of initial faith in Christ. This takes a person, who was ‘under the law’ and kills them under that law in order to raise them to new life under grace. The bible does not use ‘applicable’ or ‘not applicable’ but rather ‘under’ or ‘not under’ the law.

Before the coming of this faith, we were held in custody under the law, locked up until the faith that was to come would be revealed. So the law was our guardian until Christ came that we might be justified by faith. Now that this faith has come, we are no longer under a guardian. (Gal 3:23-25)

The Law and the Prophets were proclaimed until John. Since that time, the good news of the kingdom of God is being preached, and everyone is forcing their way into it. (Luk 16:15-20)

This is the covenant I will establish with the people of Israel after that time, declares the Lord. I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people. No longer will they teach their neighbor, or say to one another, ‘Know the Lord,’ because they will all know me, from the least of them to the greatest. For I will forgive their wickedness and will remember their sins no more. ” By calling this covenant “new,” he has made the first one obsolete; and what is obsolete and outdated will soon disappear. (Heb 8:10-13)

Naturally as a result of the Mosaic Law having disappeared (Civil, Ceremonial and Moral combined, for they can't be split in pieces) concerns about Sabbaths and Moons etc. are also way faded out:

Therefore do not let anyone judge you by what you eat or drink, or with regard to a religious festival, a New Moon celebration or a Sabbath day. These are a shadow of the things that were to come; the reality, however, is found in Christ. (Col 2:15-17)

This means the believer need not worry how to fulfill the entire law because Christ has done that for him. As a result they want to lover God and their neighbor as a result of this freedom from the curse of that law.

It is important to note freedom from the Law is only by faith in Christ so all unbelievers are are still under it. Not under the Mosaic version but the moral law of nature in the conscience that predated the version from Moses. The moral requirements and unforgiving judgment of the law within the conscience which has always existed prior to the temporary external form of it that existed for the nation of Israel, from Moses to Christ. The unbelieving Jew now, is therefore like the Gentile, now under the law of conscience that currently condemns all mankind in the same way that the Law of Moses does.

Next is the Roman Catholic view. The Roman Catholic view seems similar on the surface but is very different. It does not believe in ‘justification by faith’ the way Luther believed. It thinks of justification as merely the ‘first moment of sanctification’, so in Protestant terms is the same thing as sanctification, being a separate term only in the sequence of time. In other words, Catholic doctrine does not believe in anything called justification as understood by protestants and as such this is the division between the two main versions of Christianity, all other differences being small in comparison.

Subject of the Chapter.—Having established the conditions of justification according to the Catholic doctrine, we must proceed to say something more as to the nature of this great gift, and to show that in the justification of a sinner, his sins are wholly taken away, and that the Lutherans are in error when they assert that these sins still exist although God does not impute them to the sinner. We shall show also that when the sin is remitted, the soul also receives the great gift of habitual grace (n. 583, vi.), in virtue of which the man is interiorly renewed and thenceforth may be called by many titles expressive of friendship with God. This habitual grace is lost by grievous sin, but may be regained through the virtue of the Sacrament of Penance. “ (Outlines of Dogmatic Theology by Sylvester Jospeh Hunter, Ch 2, 634)

So under a salvation by sanctification view, a soul is maintaining satisfaction of the requirements of the Law ‘only if they continue living sanctified’. When one falls out of that state they need to confess to a priest, and get themselves back in that state again, or they may die at the wrong moment and go to hell / purgatory. So the law takes on a different meaning from the Lutheran view because it can still threaten just as though you were a sinner. In this sense you may be still under it.

The third view is a confused mixture of the two alternatives and because it is a mixture of opposing ideas it can be explained in numerous confusing ways, too numerous to count. It is this confused joining of the two previous views where ‘civil, ceremonial and moral’ splicing becomes necessary. Feeling that salvation must have always been the same and therefore the mood must always stay the same, the depressing and gloomy feeling of dread under the Old covenant is promoted to continue under the New by somehow preserving the the Law of Moses under the new. The attempt to preserve the Law is often initiated with the common expression: Did not Jesus say ‘I have not come to destroy the law’. Not realizing that fulfilling the moral law is not its destruction.

Now for the splintering opinions of specific laws that are somehow binding, like the sabbath day, or the tithe, this must be among those people who reject the main rationalized ideas of both traditional Protestantism and Catholicism. A Catholic need not ask because they may just consult the traditions. A protestant need not ask because of all the argument just made. However, if one were to imagine the moral law before Moses and ask ‘Is tithing 10% a natural law?’ or 'Is resting on a specific day an innate and natural law? It is possible one might have am affirmative opinion one way or another.

Note: As I end the answer I think it is possible I did not actually understand the question as I am unable to understand how the Law of Moses cold be understood as somehow being applicable. I will leave it up anyway as reference, in case someone does try to collect all the odd ideas people may have about some part of the Mosaic Legislation still existing and forget to include the obvious alternative that non of it exists anymore in its Mosaic form, as it was an Old Covenant during a period of Israel's history.