  In this paper, we study the birational geometry of the Quot schemes of
trivial bundles on $\mathbb{P}^1$ by constructing small $\mathbb{Q}$-factorial
modifications of the Quot schemes as suitable moduli spaces. We determine all
the models which appear in the minimal model program on the Quot schemes. As a
corollary, we show that the Quot schemes are Mori dream spaces and log Fano.
