Question is in the title.
Zero-dimensional means "has a basis of clopen sets".
Hausdorff is not enough to guarantee a countable space has dimension zero (in fact, a countable Hausdorff space can be connected).
Is regular enough?
Note 1: I assume that singletons are closed, so that regular is stronger than Hausdorff. I'm not sure what would happen here if we allow non-closed singletons...
Note 2: (countable + regular) implies normal, if that helps (?)