When will galaxies become unstable due to dark energy? When will galaxies become unstable due to dark energy? What will it be like for a civilization to observe their own galaxy going through this period of instability? 
 A: This is all pretty well described in the Wikipedia page on the "big rip".
The Friedmann acceleration equation determines the evolution of the scale factor of the universe and can be written as
 $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3P}{c^2}\right) $$
where $\rho$ is the energy density, and $P$ is the pressure, usually parameterized as $P = w\rho c^2$.
A cosmological constant has $w=-1$, which leads to
$$ \ddot{a} = \frac{8\pi G \rho a}{3}$$
And hence exponential growth. That's OK, galaxies can still survive in such a universe, held together by their self-gravity.
The "big rip" happens when $w<-1$. In this case, not only does the acceleration of the expansion increase with time, but the size of the observable universe shrinks at an increasing rate. Once this size is smaller than any particular structure, it cannot be held together. Eventually a singularity is reached when all points are infinitely separated - i.e. the scale factor becomes infinite. This occurs in a time
$$ t- t_0 = \frac{2}{3|1+w|H_0 (1 - \Omega_M)^{1/2}} \simeq \frac{11\ Gyr}{|1+w|},$$
where $\Omega_m$ and $H_0$ are the matter density and Hubble parameter now at time $t_0$.
Because of the accelerating nature of the process, the time at which galaxies would be ripped apart is very close to the end singularity (unless $w$ is very, very close to $-1$). So choose your $w$ and plug it into the formula. At present, the constraints on $w$ are not good enough to say what will happen.
