When talking about functions, there are many notions that are interpreted as well-behaved, such as increasing, continuous, integrable, and many of these terms can be directly understood in some context.
What would be equivalent notions for distributions? There is often an intuitive appeal to a notion of sufficiently random, or naturally random process, but I don't know any such notions. One requirement can be that there's a continuously differentiable definition. For continuous spaces, it can be "no set of measure $0$ has probability more than $0$". In some context, having one distribution if tails and another distribution if heads would be acceptable, and not in other contexts.
So what are the rigorous mathematical definitions that capture various notions of well-behaved distributions?