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Let $k$ be an algebraically closed field of characteristic $0$ and$\mathcal{M}_d$ the moduli space of rational maps on $\mathbb{P}^1$ of degree$d$ over $k$. This paper describes the automorphism loci $A\subset\mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus$\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groupsoccur as subgroups of the automorphism group of some $[\phi]\in\mathcal{M}_d$for a given $d$ and calculate the dimension of the locus. Next, we prove ananalogous theorem to the Rauch-Popp-Oort characterization of singular points onthe moduli scheme for curves. The results concerning these distinguished lociare used to compute the Picard and class groups of $\mathcal{M}_d,\mathcal{M}^s_d,$ and $\mathcal{M}^{ss}_d$.

Automorphism loci for the moduli space of rational maps