Nice which engine do you have, straight six or V8? Rocker covers on both are common causes of oil leaks, V8 front timing covers also.
Other common problems were pixels breaking up on radio and dash display but there are places that can repair them now by replacing capacitors.
TV display style OBC radios can also have non standard replacements now.
All E39 models are chain driven and were pretty reliable compared to later N series engines with chains made of wet pasta, in fact a lot of customers used to say they regretted trading up to the E60 in terms of driver enjoyment hence E39's, E46's and E30's are still popular choices.
As for BMW specialists were are you based?
I can highly recommend
BM Sport in Bexleyheath
Cragos in Dartford
Both have ex BMW trained staff and use genuine BMW parts