So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of.


The problem is that there exists a sequence of non-trivial Boolean locales $B_\kappa$ indexed by infinite cardinal numbers such that the only locale $X$ that has map to all the $B_\kappa$ is the empty locale

The map $B_\kappa \to 1$ are open surjections, so in particular they are effective descent map, stable regular epimorphism and triquotient map, so pretty much any class of epimorphism you might think about will contains these.

But if $X$ was projective with respect to any class containing these cover, then the unique map $X \to 1$ would lift to maps $X \to B_\kappa$ for all $\kappa$ which contradict our claim above.

Explicitely, $B_\kappa$ can be taken to be any Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$.

The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function. 

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).