The ST's are renowned for the head bearings playing up.
They may require replacement which is a bit more than I'd be happy doing personally.
If you check them as Keith suggests and there is no noticeable graunchyness ( copyright Barnard Dictionaries
) then I'd try setting them.
There is no C spaner used on the ST range.
To adjust the head bearings, loosen the pinch bolt in the top yoke holding the split nut at the top of the steering head.
Using the special tool to locate into the holes on the split nut, set the preload on the bearings at 30Nm of torque.
Either that or drop in to your favuorite cross dressing Ducati specialist & it takes 5 minutes to do.
Mine was vibrating & shaking like a b@rsteward until I set them as above.
It's splendid now!