We do not know how many things are not believed. We do not know how many things are not regretted. We can assume that there are more than one things not regretted or believed. This is because any suggests more than one. But it could be any number larger than one without a limit. It is whatever amount of actions that have been committed by the speaker.

I belive/regret none of those.

It's all about the negative. I don't believe any of it. = I believe none of it. I don't regret any of it. = I regret none of it.