My 97 XK8 coupe rides smoothly at low speed, about 50km or so. Once the speed picks up, there's a shuddering vibration that seems to come from the rear. Otherwise it tracks straight. It gets a little stronger as speed picks up, but doesn't increase in frequency. The car is new to me, so I don't know if it has been doing this for long.
It also has the dreaded abs fault, though I don't think it is related.
Any ideas?

 

Check the rear wheel bearings. The rear ABS can go off with vibration in the bearings.
If good the U joints are something that will also give a higher speed vibration.

 

olphart,
Well obviously the vibration is coming from imbalance of a motive element within the rear drive train. Some causes can be internal and others external. It is easy to inspect the external first and tick them off a list: Ideally, jack the whole rear of the car up so that you can rotate the wheels individually and check carefully.
Tyres - debris / damage?
Wheels - Debris / damage / balancing weights missing?
Brake rotor disks - Warp / damage?
Drive half-Shafts - damage / Tangled foreign body?
Universal Joints - Wear / damage?
Prop shaft - damage / Tangled foreign body?
Prop shaft end Joints - damage / wear?

With the above causes eliminated, the rest of the potential causes are internal and more difficult to inspect.
Wheels Bearings- Wear / Dry?
Differential - damage / Wear?
Gearbox - damage / wear?

 

So far, the wheel bearings and brakes are ok, so we're down to checking out the u joints.

 

I have read on other post that this might happen with a bad wheel balance. The consensus was to get the wheels hunter road force balanced not just a normal balance.

 

Spoke too soon, front right wheel bearing is bad. Hand brake doesn't work. Need to find a new cable for that