How well can we measure how fast are we spinning? Although absolute translational motion is meaningless and unmeasurable (Michelson and Morley, etc), absolute rotational motion is meaningful (Newton's bucket) and measurable, using Foucault's pendulum. In 1851 Foucault showed this could be used to measure the earths rotation, $1 \over 4$ degree per minute.
What can we achieve using the improvements of modern technology? After 170 years of development, could we build an experiment that, without any external observations, measured the rotation of the earth round the sun? And the rotation of the sun round the galaxy? And even (this is stretching it, but interesting) whether the whole universe has some rotation?
 A: Direct, instantaneous monitoring of the Earth rotation rate is possible with a large ring laser interferometer, secured to bedrock.
In Germany there is a facility 'Fundamentalstation Wettzell'. This facility operates a setup called 'Ring Laser G'.
Ring laser G is a ring laser setup with the mirrors as the corners of a 4x4 meter square.
In a ring laser setup clockwise propagating light keeps propagating clockwise, counter-clockwise propagating light keeps propagating counter-clockwise, that is how the mirrors are set up.
If there would be absolutely perfect reflection the clockwise and counterclockwise propagating light would never interact. However, while the mirrors are 99.9999 percent efficient, the remaining backscatter tends to keep the two counter-propagating beams of light locked to the same frequency.
When the ring laser is sufficiently large even the slow Earth rotation rate is enough to unlock the two beams.
The ring laser is anchored to bedrock. Due to the rotation of the Earth a frequency difference arises. (During the time that the light goes around the source moves, so the clockwise and counter-clockwise beams don't travel the same length) The magnitude of the frequency difference is measured by allowing some of the light to exit and then obtaining interference between the two beams. The resulting interference pattern is a beat frequency. In the case of the Ring Laser G this beat frequency is arond 348.6Hz.
Fundamental to the operation of a ring laser is that establishing the point of zero rotation does not require calibration. When a ring laser is not rotating there is no frequency shift, hence no beat frequency.
(Of course there are practical difficulties, such as the already mentioned tendency of the beams to remain locked.)
Given the dimensions of a setup the expected beat frequency can be calculated in advance. So even in the absence of any other data the Earth rotation rate can be inferred from the magnitude of the observed beat frequency.
The operating principle of ring laser interferometry (and other forms of ring interferometry) is the Sagnac effect.
With a ring laser interferometer you observe whether you are rotating with respect to inertial space.
A ring laser gyro device is the optical counterpart of a gyroscope. As we know: a spinning gyroscope, when perfectly undisturbed, remains in the same orientation with respect to inertial space.


The web page lists among facility's goals:
Detection of short-term spin fluctuations with a resolution of $10^{-9}$
Detection of short-term polar motions with a resolution of 0.2 mas or 6 mm
Near real time acquisition with a temporal resolution of 1 hour or less
The particular page with that information was last updated in 2005. I cannot find whether that setup is till running, or whether it has been shut down.


The physics department of the University of Canterbury New Zealand was leading in the development of ring lasers for Earth monitoring. Their facilty (including a ring laser setup far larger than the one at Wettzell) was located in caverns near the city of Canterbury. As far as I know the Earthquake in Canterbury has shut down those activities.
A: The earth rotates on its axis at a rate of roughly $360$ degrees in $24$ hours, which is $15$ seconds of arc per second. The plane of a Foucault pendulum at the North or South pole would precess at this rate. The precession rate of a pendulum at other latitudes is smaller by a factor of the sine of the latitude.
The angular speed of the earth in its orbit around the sun is $\frac 1 {365}$ of this rate, which is about one arc second every $24$ seconds. The orbital period of the solar system around the centre of the Milky Way is between $225$ and $250$ million years. This corresponds to an angular speed of approximately one arc second every $200$ years.
It is conceivable that a very precise version of a Foucault pendulum (or an equivalent system such as a gyroscope) could detect the small difference in precession rate caused by the earth's orbit around the sun - I don't know if this has ever been attempted. Detecting the solar system's orbital motion around the Milky Way in this way is not even remotely practical.
As to detecting the rotation of the whole universe - what would it be rotating relative to ?
A: 
...absolute rotational motion is meaningful...

I understand the word absolute here in the same meaning that we say: acceleration is not relative in SR. An accelerated frame knows that it is accelerating, and can measure its value by dropping objects for example, and measuring its local acceleration.
In the case of an accelerated frame, it is possible to change coordinates and return to Minkowski spacetime. Because that is exactly the frame of the falling object.
All that proofs of earth rotation belongs to the same framework in my view. After calculating interference pattern of the Ring Laser G (as mentioned by Cleonis), or the turn of the plane of swing in the Foucalt Pendulum, it is always possible to change our frame of reference to the frame where there is no interference, or the frame of the swing plane of the FP. That is a non rotational frame.
If now we compare with a gravitational field, it is not possible to get the Minkowski spacetime by changing frames. It is not possible to get rid (completely) of the tidal forces.
I wonder if the idea of a really absolute rotation is  something where this mental operation of moving to another frame from where our own frame is rotating is no longer possible. Some tiny effect, like tidal forces for gravity, that would be an indication of a rotation of the whole universe.
