Define noticeable. Gravity Probe B just took very accurate readings of the frame dragging effect of the Earth. We've known about the frame dragging effects of the Sun for about a century now, as it was used to explain the precession of Mercury around the sun being higher than Newtonian physics allowed.
I'm not sure what your asking about in the last half of your post. Frame dragging and spacetime warping are purely determined by the mass of something. There isn't anything that we could feasibly build at this point in time massive enough to detect frame dragging from it.