In the presence of pullbacks, the intersection of subobjects is given by their pullback. Whenever image factorization exist, the union of subobjects is given by the image of their coproduct. Sometimes, e.g in abelian categories and topoi, binary unions are effective - the union of two subobjects is the pushout of the obvious span over their intersection.
I was wondering - are binary unions of regular monomorphisms effective in any regular category?
No. Let $\mathscr{C}$ be the category of monoids and let $X$ be a non-trivial monoid. Let $(X\times X,p_1,p_2)$ be the product of $X$ with itself and let $i_1,i_2 : X\to X\times X$ be the unique morphisms such that $p_1 i _1 = p_2 i_2 = 1_X$ and $p_1 i_2 = p_2 i_1 = 0$ (i.e. the morphism sending everything to the identity element). Writing $0$ for the trivial monoid we see that the diagram $$\require{AMScd} \begin{CD} 0 @>>> X\\ @V_{}VV @VV{i_2}V \\ X @>>{i_1}> X\times X \end{CD}$$ is a pullback. However the above diagram is not a pushout, since products and coproducts of monoids do not coincide. This answers the above question since $i_1$ and $i_2$ being split monos are certainly regular.