Both the E39 5 series and the E38 7-Series and other models 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 & E39 and other models have 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.
Wobbles that aren't shimmies
Let's first get rid of the wheel-wobble that isn't shimmy. If you get a wheel-wobble that really feels as if the wheel is coming loose and gets worse when the brakes are used then the first thing to suspect is a sticky brake calliper. The difference between shimmy and the wobble caused by a sticking brake calliper is that latter causes a massive wobble, the steering wheel is difficult to keep hold of and the wobble keeps going when you slow down, almost to a standstill.
Vibration that only shows itself when giving the car some welly, especially in 2nd and 3rd gear is quite often due to the propshaft centre bearing. The bearing is held in a rubber-enclosed housing. When the bearing starts to fail the rubber soon deteriorates and this allows the propshaft too much lateral movement. The movement causes a vibration through the car, it will eventually get to a point where the propshaft will start thumping the transmission tunnel. This type of problem always goes away when the throttle is lifted whereas shimmy doesn't.
Getting rid of the shimmies - the easy things
Wheels
The best place to start (and often the cheapest) is with the road wheels. The first thing to check is if the wheels are right for the car.
If you have fitted an alloy from a model with a different centre bore then you must fit spigot-rings. The only problem is that spigot-rings are not normally high-precision parts, they wear and corrode and when they do they will not hold the wheel central to the hub. If you are using spigot-rings consider getting alloys with the correct offset and bore, or, at the very least get new spigot rings.
Even if you are using standard wheels or are using wheels with the correct bore and offset it is worth cleaning the bore and mating surfaces on both the inside of the wheel and the hub. Where there is corrosion on the hub or wheel the wheel will not mate correctly with the hub.
Tyres
Cheap tyres are often a cause of problems, although these balance fine at low speeds, once they are taking the load of the car and are rotating at speed they do not retain their shape or were not round in the first place. There is only so much that sticking lead weights on the rim can compensate for. If you have a lot of lead on your wheels then something is not right in the first place. If you do not have staggered wheels fitted it is well worth rotating the wheels from the back to the front of the car to see if there is any difference. If it makes a big difference it is probably a problem with the tyres.
Getting rid of the shimmies - the harder bits
Once all the easy options are out of the way we have to move on to the front suspension components. There is a lot of advice given on what to change first but in my experience most problems are either removed entirely or at least made significantly better by replacing the upper arms and rubber bush.
While you are at it you might as well replace the lower arms as well , the cost is not all that high and is worth doing once the car is on the lift. A big difference can be made by just changing the bushes but normally if the bushes are damaged then the ball-joints are damaged at the same time. Some like to replace the bushes with polyurethane after-market versions, be aware though that this will significantly affect the quality of the ride and may well lead to other component failures. The bushes have to be pressed into the arms if you buy them separately.
Getting rid of shimmies - it's getting much harder now
We have now covered the possible causes in 95% of the cases I've been involved with. I have heard of miracle fixes where just one component of the front suspension has been replaced before the usual things have been tried but these are really just one-offs. So, in order of the most likely here are the remaining candidates:
Front stabiliser joints
Centre tie-rod , Drag link bearing and Tie rods
Once you have got to this point things have got pretty expensive. You can get complete suspension refurbishment kits from Ebay.de, I have never tried these but have heard good things about them. My present E38 735i had shimmy which was cured by replacing the centre tie rod. However, the giveaway for a centre tie rod failure is that they groan when the engine gets hot, and can be expensive!