We study the directed polymer model in dimension $1+1$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $\beta=\beta_n$ vanishes as the size of the system n goes to infinity. When $\alpha\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n}\le h_n\le n$ can be achieved by tuning properly $\beta_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when $\alpha<1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP].