Since the original question was, in my reading, about things which are NOT prohibited by law but ARE prohibited by morality, I'll address that. It seems that most posters are addressing the opposite, activities that are morally neutral or good, but prohibited by law.
There are hundreds of things which are immoral but not illegal. It's hard to list them all. For one thing, different things are illegal in different states/countries/Federation planets. For another thing, I would say there are two categories within this set: Things that are immoral and SHOULD be illegal, because they are harmful to society or individuals, and things which are immoral and SHOULD NOT be illegal, primarily because to prosecute them would be sufficiently onerous as to approach absurdity.
As an example of the second, I put forward lying. Lying is immoral, but in most circumstances, not illegal. But it is so common and would be so difficult to catch/prove, that greater evil would be done attempting to criminalize it than leaving it "legal". Similarly, many forms of sexual immorality, most of which I won't go into for fear that the internet flames will sear my very eyes, and I shall never open them again save to weep.
Speaking of saying things that will get me flamed: abortion is an example of the first category. Most of the world legalizes the practice of killing an unborn human, many places with restrictions, others without. While most here would probably disagree with me in general, consider that. In many places, including my own country and state, a medically viable child can be killed. That is immoral. If that's still not heinous enough to prick your seared conscience, consider that, in past ages, flat out infanticide was not only legal, but sometimes compulsory.
Or adultery. Now this one is often illegal on the books, but almost never enforced in the West. Again, many people have different sexual ethics, but I contend that when you promise to remain with and bond with one person, and then the state recognizes that promise and confers benefits upon you based on that promise, then the state has a legitimate interest in punishing your breach of that promise.
Anyway, that should be enough to get me in trouble.
And now, a bit of fun...
Kair said:
Humans are treated as animals and are trained as such, a symptom of the fact that the vast majority of people exist in a twilight between human and animal....humans are seen as animals...heavily animalized humans...other animalized humans...more towards human than animal. It is hard to save the world, and I can't do it alone.
I hold at your neck the gom jabbar, the high-handed enemy!
Keep it up, comrade. The Kwisatz Haderach cannot be far away!