Long, 

For the $\mathbb{Q}$-Gorenstein question, I don't know of a reference either but it should be easy, if $\omega_R^{(n)}$ is locally free at a point, then it's locally free in a neighborhood of that point (of course, I'm probably assuming normal or G1 + S2 to make sense of $\omega_R^{(n)}$).   

A couple other that jump to mind are the following.

I.  Seminormality / weak normality.  

II.  Being $F$-split in characteristic $p$ (at least in the $F$-finite case, $F$-finite is another decent one on its own).  Of course, all the sings of the MMP (canonical, terminal, lc, klt, slc, rational, Du Bois, etc.)  Most of the singularities of tight closure theory (strong F-regularity, F-purity, F-injectivity, F-rationality, some of these requiring again the $F$-finite case). 

With reguards to your second question:  The general common thing that virtually all the singulraity classes mentioned above possess, which makes them open is the following:

Almost all of these singularities are checked by either showing that a particular module $M$ is isomorphic to $R$ or $\omega_R$ is that a particular module is zero.  For example, klt and log canonical singularities can both be checked by this (is the multiplier ideal / non-lc ideal equal to $R$).  This also holds in the characteristic $p$ world, although its not the usual way things are phrased.  

For example, strong $F$-regularity can be checked by looking at the ideal 
$$
J = \sum_{e \geq 0} \sum_{\phi} \phi(F^e_* cR)
$$
where $\phi$ runs over all elements of $Hom_R(F^e_* R, R)$ and $c$ is chosen such that $R_c$ is regular.  If this ideal equals $R$, then $R$ is strongly $F$-regular.