Assume $X$ be a normal projective variety with $\mathbb Q$-Cartier divisor $D$, then can we extend adjunction formula on pair $(X,D)$?
If the pair $(X,D)$ is log canonical and $S$ be a component of $D$ with coefficient 1, then we have adjunction type formula as $$K_S+D_S=(K_X+D)|_S$$
[edited]: If $f:Z\to X$ be a finite Galois map, then there exists a branch divisor $B$ on $X$ s.t, $K_X+B$ is $\mathbb Q$-Cartier and $K_Z=f^*(K_X+B)$.
As an example on surfaces: Let $(X,D)$ be a surface with Kawamata log terminal singularities and $\pi:Y\to X$ be a minimal resolution of $X$. Then we can write $\pi^*(K_X+D)=K_Y+D_Y$ where $D_Y\geq 0$ and each exceptional curve of $\pi$ is a smooth rational curve.
Note: If $K_X+B$ is not $\mathbb Q$-Cartier, it is not clear what the adjunction formula should mean, but even then one can have a sort of adjunction formula involving $Ext$'s which is Grothendieck Duality.
See Kawamata-Kodaira canonical bundle formula also
Example: Let $f:V\to X$ is a contraction but not birational such that $K_V\sim_\mathbb Q 0$ (like Calabi-Yau model)then it often happen that $K_V\sim_\mathbb Q f^*(K_X+B)$ for some divisor $B$