Let $ U=U_q(g) $ be the quantized enveloping algebra with a triangular decomposition $U = U^-U^0U^+$. We classify all $U$-module structures on $U^0$ with the regular action of $U^0$ on itself. We obtain that the necessary and sufficient condition for the existence of such modules is that $U$ has to be of type $A_n(n \geq 1)$, $B_n(n \geq 2)$ or $C_n(n \geq 3)$. We also study their module structures and associated weight modules for each type.