Was the filling of the Three Gorges Dam's impact on the Earth's rotation rate detectable?
The .06 microseconds figure doesn't sound accurate

The moment of inertia for a solid sphere is $$I = \frac{2mr^2}{5},$$ where $m$ is the mass of the sphere and $r$ is its radius. Applied to the Earth, this gives us $$I = \frac{2\cdot5.972\times10^{24} \text{ kg} (6371000 \text{ m})^2}{5} = 9.70\times10^{37} \text{ km-m}^2.$$

A simplified estimate of moment of inertia for the water in the three gorges dam is $mr^2$. The change in moment of inertia from raising that water by 175m can be calculated by $$\begin{align}\Delta I &amp;= m_{water}\left[(r_{earth}+175)^2-r_{earth}^2\right] \\ &amp;= 39\times10^{12}\left(6371175^2-6371000^2\right) \\&amp;= 8.69\times10^{22} \text{ km-m}^2.\end{align}$$

Assuming that the angular momentum ($L$) of the Earth stays constant, then since $L=I\omega$, $$I_{before}\omega_{before} = I_{after}\omega_{after},$$ or $$\omega_{after}=\omega_{before}\frac{I_{before}}{I_{after}}.$$ The angular velocity, $\omega$, of the Earth (before) is $7.292\times10^{-5} \text{ radians/s}$. The discrepancy between the before and after moment of inertia is so insignificant that I would be wary of floating point error on this calculation, but using Python2 I calculate the delta in rotational speed to be $6.776\times10^{-20} \text{radians/s}$. For a rotation of the Earth this means the rotation will take about 80 picoseconds longer, or 0.00008 milliseconds. 

Conclusion

The statements made by VSauce don't match up with Physics 101. I am not inclined to believe this claim unless there is some other explanation or compelling evidence. Given that the magnitude of the change in rotational speed is three orders of magnitude less than he claims, and that 80 picoseconds is about 1/4 of the time it would take your computer to add 2 integers, I think it is safe to say that this is not measurable.