Since your question is kind of vague, I'll answer the question that I think you're asking:

The theory of generalized spaces usually goes a bit beyond locales, namely to Grothendieck topoi. Locales are a reflective subcategory of Grothendieck topoi, so you can think of Grothendieck topoi as a true generalization of locales. There are many examples of topoi that don't have enough points that occur frequently in geometry, for example, the topoi of sheaves of sets on the Étale and Nisnevich sites.

I am not a logician, but I think that the examples you're looking for are all usually stated in this language of topoi. Proving that these topoi are the sheaves on a localic site is usually how people think of such things.

Reference: https://ncatlab.org/nlab/show/locale#RelationToToposes

This is covered in great detail in Johnstone's Sketches of an elephant

Since your question is kind of vague, I'll answer the question that I think you're asking:

The theory of generalized spaces usually goes a bit beyond locales, namely to Grothendieck topoi. Locales are a reflective subcategory of Grothendieck topoi, so you can think of Grothendieck topoi as a true generalization of locales. There are many examples of topoi that don't arise from topological spaces that occur frequently in geometry, for example, the topoi of sheaves of sets on the Étale and Nisnevich sites.

I am not a logician, but I think that the examples you're looking for are all usually stated in this language of topoi. Proving that these topoi are the sheaves on a localic site is usually how people think of such things.

Reference: https://ncatlab.org/nlab/show/locale#RelationToToposes

This is covered in great detail in Johnstone's Sketches of an elephant