Is it easy to construct examples of smooth complex projective surfaces $S$ of general type such that $h^0(\Omega\_S)>1$, $alb\_S:S\rightarrow Alb(S)$ is generically finite (unto its image) and
$${\rm Bs}(\Omega\_S):=\{x\in S,\ \omega\_x(T\_xS)=0 \ \forall \omega\in H^0(\Omega\_S)\}$$
is non-empty?



Base locus of cotangent bundle of a surface
I am just posting my comments as one answer.


Let $A$ be an Abelian variety of dimension three. Denote by $\Theta$ an ample divisor class on $A$. Let $S’$ be a smooth effective Cartier divisor on $A$ in the algebraic equivalence class of $n\Theta$ for very positive $n$. Let $f:S\to S’$ be the blowing up of $S’$ at a point $p$, precomposed with a blowing up of a point on the exceptional divisor over $p$. Denote the exceptional locus of $f$ by $E$. It is a union of two rational curves that intersect at a point $q$.


By direct computation, the pullback map is an isomorphism from $\Omega\_{S’}|\_p$ to $H^0(E,\Omega\_S|\_E)$. Thus every $1$-form on $S$ agrees on $E$ with the pullback of a unique $1$-form on $S’$. The difference is a $1$-form that is zero along $E$. This is the same as a global section of the coherent sheaf $\Omega\_{S’}$ that vanishes at $p$.


By Serre vanishing and duality, both $h^1(A,\mathcal{O}(-S’))$ and $h^2(A,\mathcal{O}(-2S’))$ equal zero. Thus, the conormal sheaf of $S’$ also has vanishing $h^1$. Thus, by the fundamental short exact sequence, every global section of $\Omega\_{S’}$ lifts uniquely to a global section on $S’$ of $\Omega\_A|\_{S’}$. Because $h^1(A,\Omega\_A(-S’))$ is zero, this lifts further to a unique global section on $A$ of $\Omega\_A$. Of course the only global section of this free $\mathcal{O}\_A$-module that vanishes at $p$ is the zero section. Altogether, every global $1$-form on $S$ is the pullback of a unique global $1$-form on $A$. Of course all of these vanish at $q$.


