Stern implies uncompromising, inflexible firmness, and sometimes a hard, forbidding, or withdrawn aspect or nature: a stern parent. Severe implies strictness, lack of sympathy, and a tendency to impose a hard discipline on others: a severe judge.

Strict means you must follow the rules, without any exceptions. Stern means that you are serious about disciplining people who break the rules. Rigid just means stiff, but when describing a person it means they aren't willing to change. It's a very small difference.

Strict and stern are pretty much the same, but strict would more likely be applied to rules or actions while stern would more likely be describing someone. Severe would describe how bad something is which could apply to many things. "My school rules are very strict."