When are all monos regular?

This might be an embarrassingly easy question, but I always get confused about it. When are all monos in an algebraic category regular (or more generally, when are all monos in a regular category regular)? What are some sensible sufficient or necessary conditions? For example, all monos in the category of groups are regular. How about the category of lattices, or lattices with a top element? Andrej Bauer