Both the E32 and the E38 7-Series can suffer from the dreaded 'shimmy'. This is wheel-wobble that usually first shows itself under braking, for instance, when coming off a motorway intersection and braking from high speed. The wobble can be felt through the steering-wheel but it usually disappears as the car speed reduces.
Sometimes this is all you ever get, but more usually you will start to get wobble as the car is accelerated through 50 - 60MPH. The more you drive the car the worse it will get. The wobble is quite often accompanied by tram lining and a general feeling of imprecision from the steering in general.
The shimmies are often an indication of general wear in the steering system and suspension, the BMW E38 has a great deal of weight over the front wheels, especially when braking. It also seems to have a resonant frequency that occurs between 50 and 60 MPH, if there is any weakness it is at this speed where the problems will occur.
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