How does one research morality?

Mostly in surveys, but sometimes in experiments as well.

Honestly, Alex, there are lots of scientific papers on that topic, but not in the sense you obviously mean. I completely agree that what is right or wrong is not a question for science (I agree with everything after your first line).
However, which people find which acts moral, whether there is a common hierarchy of morality across cultures or not, how the feeling of morality develops with age, in which way religious conviction has an influence on opinions on moral, and many more questions that are at least in principle amenable to an empirical answers have, of course, been studied by social scientists.

Wolfgang