Mechanical gyroscopes
For an observer at the surface of the Earth, the Coriolis force would be responsible of moving the gyroscope axis of rotation. It is well-known that this force is weak on small system, by which I mean it is easily overwhelmed by non-inertial forces such as friction, viscosity, etc. This is why Foucault pendulums have to be sizeable (the smallest I am aware of has a string about 5 meters long). Thus I reckon you would need a gyroscope way bigger than the mass-produced ones. I am not aware of any purpose-built gyroscope to test Coriolis forces. But why would anybody bother when it is so much easier to build a Foucault pendulum?
Laser gyroscopes
Mechanical gyroscopes have been completely replaced by laser ones in all inertial system of planes, rockets, etc. They are based on the optical brother of the Coriolis force, called the Sagnac effect. The principle is rather simple and can be demonstrated on the following simple sketch (kudos to Krishnavedala on Wikipedia: I just converted the image to PNG)
So a light source (nowadays a laser) sends two beams across the optical loop (the beam is reflected by the mirrors, represented by the grey rectangles): one in one direction and the other one in the other direction. Then the two beams meet at the detectors, which just records a shift $\Delta$ of interference fringes between them. This difference is equal to
$$\Delta = \frac{4A}{\lambda c}\omega\sin\phi$$
where $A$ is the area enclosed by the loop, $\lambda$ is the wavelength of the light, $c$ is the speed of light in vacuum, $\omega$ is the angular velocity of Earth, and $\phi$ is the latitude.
So again, the effects is bigger for a bigger detector, and commercially available laser gyroscope have a size of the order of 10 cm, and that gives a fringe shift several orders of magnitude too small to be measurable.
But the experience has been done with a very large ring, whose perimeter was about 2 km (or 1.2 miles), by Michelson, Gale and Pearson in 1925 (the Michelson of the other famous experiment). The measured shift was 230 parts in 1000 for an accuracy of 5 parts in 1000, as compared to a predicted shift of 237 parts in 1000, with the above formula. So clearly Michelson et al did measure the rotation of Earth. The Wikipedia article has more details, including the references to the three original papers of Michelson et al.
With the advent of lasers, and modern instrumentations, the precision of laser rings has vastly increased, for much smaller rings. For example the C-I ring in Christchurch, New Zealand, has a perimeter of only 3.477 m but it has a sensitivity limit of 20 microarcs per seconds for a collection time of one hour. The Christchurch team (website) wrote a nice paper describing not only all their wonderful devices but also giving an in-depth historical, theoretical and experimental review of laser rings [a].
In that paper, I learned that some industrial laser ring using fibre optics can actually sense Earth rotation. Quoting Stedman et al:
Fibre-optic gyros have proved useful in a very wide variety of fields, such as
oil prospecting. The Sagnac signal generated by the Earth’s rotation is
sensitive to the orientation of the device, and if the orientation is defined by
the ring laser being totally embedded in an oil-well drill bit, the alignment of
exploratory bores can be monitored deep underground. Thanks to advanced
technological tricks including dither, such small gyros monitor the projection
of the Earth’s rotation to an accuracy of $0.01^\circ h^{−1}$
or $5\times10^{−8} rad s^{−1}$, a fraction $7 \times 10^{−4}$ of the Earth’s
rotation rate $\Omega_E$ (Kim and Shaw 1986).
The given reference is [b].
[a] G E Stedman, Ring-laser tests of fundamental physics and geophysics, Reports on Progress in Physics 60 (1997), 615
[b] B. Y. Kim and H. J. Shaw, Fiber-optic gyroscopes: in harsh, confining environments this advanced gyroscope, a close cousin to the ring laser gyro, offers great advantages, IEEE Spectrum 23 (1986), 54-60
