Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the function
$$\dim H^0(X_p, m K_{X/D}~_{|X_p})$$ is constant on $D$ (where now $X_p$ denotes the fibre over the point $p \in D$ and $D$ is a disc).
1) I am lokking for examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical.
2) Is there any approach for the following conjecture?
Conjecture: Every variety with a single pluricanonical form is birational to a variety with canonical singularities such that in addition the canonical divisor is nef