Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to find any reference. 



Properties inherited by the coarse moduli space
Not at all. Take for instance the quotient stack of $\mathbb{A}^2$ by $\pm 1$. The coarse moduli space is the quadratic cone.


