Thanks for your question.
The tricky part about answering this question is that we generally consider there to be a certain morality that automatically attaches to following the law. In general, the argument is that the rule of law is a necessary (or at the least, a beneficial) thing for humanity as a whole, given that we are social creatures and must live with each other. Thus, breaking a law, no matter what it is, carries some sense of immorality, since it weakens that structure we all live within. To be technical, therefore, we should require that our illegal acts be not merely morally neutral, but that they should have enough moral value to outweigh the moral costs of illegality.
The first, and most important answer is civil disobedience –the breaking of a law that is itself immoral and unjust. Paradigmatic examples from the recent past include violations of the laws of segregation in the American South or the laws of apartheid in South Africa. Illegal strikes and protests can also come under this categorization, when they stand in opposition to practices that are cruelly exploitative or harmful.
If we set aside the objection that breaking the law is immoral in itself, there are many practices which are illegal, but are arguably not immoral in of themselves. For example, to drink (any) alcohol at age eighteen is illegal in the United States, yet (unless you believe alcohol drinking to be intrinsically immoral) it is not immoral outside of its illegality. The putative justification for the law is that eighteen-year-olds are not mature enough to drink safely and responsibly –if eighteen-year-olds drank exclusively in moderation, the law would lose its moral justification.
A final category of illegal-but-not-immoral actions is the breaking of laws which are themselves ridiculous or meaningless. For example, the internet tells me that Idaho state law makes it illegal for a man to give his sweetheart a box of candy weighing less than fifty pounds. Laws of this nature are generally ignored by common consent. The argument here is that actually following such laws would be of greater damage to the rule of law than to break them.