Call a finite dimensional selfinjective algebra $A$ nice in case $Ext^{1}(X,Y) \neq 0$ (or equivalently $\underline{Hom}(X,Y) \neq 0$) for arbitrary indecomposable modules $X,Y$. Is there a classificaiton of nice selfinjective algebras? I think the only examples might be $K[x]/(x^n)$, but Jeremey Rickard found another example in his answer.