Surely:
i] gravitomagnetism is a "magnetic equivalent", and therefore, like magnetism itself, can only affect moving objects, and so is not responsible for stationary objects rotating?

ii] gravitomagnetism is simply the covariantly obvious fact that a purely linear (ordinary) "force" (whether gravitational or not) in one frame will be partly "torquey" in another non-rotating frame, and therefore has nothing to do with general relativity, and particularly with frame-dragging, or with rotating frames?

The math at the bottom of the frame dragging article surely implies it's a relativistic effect with curved space and proper time considerations.

In addition, I believe the relativistic insights on frame dragging/rotational effects came from the einstein curvature tensor because the polarization of a gravitational wave is quite different from that of a magnetic wave....

There are some really fantastic 3 dimensional graphics online I saw recently....I can't find them, but will post here if I do...they more clearly illustrate the effects...anybody know where to find them, they were in almost irridescent brillant blue...

Both of your Wikipedia references look correct to me, and your suggestions above both look incorrect.

i] Magnetism doesn't only affect moving objects; it also affects objects with a magnetic dipole moment (such as other magnets). In the gravitomagnetic model, the gravitomagnetic field is an angular velocity and the equivalent of the magnetic dipole moment is angular momentum.

ii]If gravitomagnetism is treated like electromagnetism in special relativity, then one of its effects is to correct the Lorentz transformation of the gravitational acceleration when viewed from other frames, in the same way as magnetism and electricity. However, this approximation is not very useful, as it assumes flat space but for gravity the curvature of space is essential to an accurate relativistic picture, and a more accurate calculation turns out to need a factor of 4 (or various factors of 2 and 1/2) to give the right result.

Yes, but the gravitational equivalent of a magnetic dipole moment is ordinary spin … and so planets of opposite spin, moving in the same orbit through the "gravitational field" of a star would experience gravitomagnetism in opposite directions (and along their spin-axes!).

and this does not depend on the rotation of the star, but only on the standard Lorentz transformation.

A rotating star exerts an additional force to gravitomagnetism, which is the same for both planets … and that is frame-dragging.

My objection to wikipedia's …

… is that frame-dragging is a "field effect caused by moving matter", but isn't gravitomagnetism.

and to wikipedia's …

… is that the main consequence (of gravitomagnetism) is that its path will curve slightly

… a free-falling object will not be made to rotate by gravitomagnetism … gravitomagnetism cannot alter the rotation of an object (it does no work!)

You are apparently using the term "gravitomagnetism" in a different way to the conventional way. The gravitomagnetic field is locally equivalent to a rotating frame of reference, in exactly the same way that the gravitational field is locally equivalent to an accelerating frame of reference.

In a first-order approximation, the gravitomagnetic field does indeed cause a deflection of a moving mass proportional to the velocity, in the same way as a magnetic field deflects a moving charge. However, it is the same field which is also responsible for the rotation effects in objects at rest or in free fall.

Relative to a fixed frame, gravitomagnetism can do work if it acts on an object which has angular momentum in a direction which is not aligned with the field, in that it causes an effective torque proportional to the cross product of the field and the angular momentum, and that torque can do a fixed amount of work aligning the angular momentum with the field. From the point of view of the object in the field, no work is done (in the same way as gravitational acceleration does no work on a free falling object as seen from its own point of view).

If the object has no angular momentum, a constant gravitomagnetic field does no work on it and does not induce rotation, but an observer at rest relative to the object will feel as if it is rotating. Changes in gravitomagnetic field can induce changes in rotation.

tiny-tim, Gravitomagnetism and Frame-Dragging are two different different analogies to visualize the effects that occur at the vicinity of mass currents.
They both apply, but IN DIFFERENT REGIMES.
Take the example of a Kerr Black Hole. The gravitomagnetic picture (i.e., the analogy with a magnetic field) applies far from the black hole, where the non-linearities of the gravitational field are negligible, and the equation for the geodesics takes a form similar to the electromagnetic Lorentz force for a charged particle subject to an electromagnetic field.
In the far field limit, it is correct what you say: only moving test particles will be affected by the gravitomagnetic field of the source. Non-spinning test particles at rest will not be deflected. Just like in electromagnetism.

If you get closer to the black hole, however, this will no longer be true. Close to the black hole, the gravitomagnetic picture no longer holds because the gravitational field is highly non-linear.
That is the regime where the picture of frame-dragging (by analogy with fluid-dragging) comes into play: all particles are dragged, irrespective of their motion. And, as you probably know, when you reach the ergosphere, all particles (even photons) are forced to co-rotate with the black hole, irrespective of their initial velocities.

Actually, the linear gravito-electromagnetic analogy is non-covariant, thus it can never lead to the correct transformation of gravitational fields (and it is not only for those factors of 4 or 2). Such analogy holds only for the so-called “static observers” (for a discussion on the subject, see http://arxiv.org/abs/gr-qc/0612140, sections 2.2, 2.2.1 and 3).

Thanks - that looks like an interesting paper. From previously working out the laws of motion in isotropic coordinates near a central mass then transforming that by Special Relativity for certain special cases I was already aware the gravitoelectromagnetic equivalent of the Lorentz Force law is missing an important term which related to the tensor nature of gravity, but that paper looks as if it handles this in a more general way.

Loosely speaking, where electromagnetism has a four-vector factor (c,v), gravity has a factor like (1/c)(c,v)^{2}, equivalent to (c(1+v^2/c^2),2v). What's more, the coordinate value of c varies with potential, and a boosted isotropic coordinate system isn't isotropic, so it all gets very messy.