As far as I can tell, you're looking for open conditions, and you should therefore expect them to look kind of silly when you try to check over points. Standard examples:

1. asking for something (like a function) to be non-vanishing.
2. asking for smoothness in a flat family.
3. representability of a moduli problem

In some cases, you're looking for open conditions, and you should therefore expect them to look kind of silly when you try to check over points. Others seem to make use of "semicontinuity plus quasi-compact badness" theorems. Standard examples:

1. asking for something (like a function) to be non-vanishing.
2. smoothness (from generic flatness).
3. morphisms having fiber dimension at least n.
4. representability of a moduli problem.

For some of these, we have to be careful calling them open conditions because of non-faithful behavior like empty fibers. We can't always define the empty case to be bad, e.g., the empty scheme is smooth. (Is "smooth + nonempty geometric fibers" an open condition?)