Macarto not sure what you are looking for by way of 'amenities'. Most apartment blocks rented by expats (other than those from the 3rd world) will have a gym and a pool. Downtown, as others have said, is around the Burg. Handy for the Metro and Fibber McGees and McGettigans (Irish style 'pubs'). Dubai Mall has all the shops you could ever need incl Waitrose + ice rink, multi-screen cinemas etc.
Business Bay is more or less next door to Downtown and a little bit quieter but might suit. No major shopping malls but a few choice watering holes. Try the Eloquent Elephant in the Taj Hotel.
The Marina is quite a bit further from DXB ( but closer to the 'new' airport if that matters. ) Not the best spot for car owners because of traffic and the tower blocks are too close together for my comfort - given how often they've become towering infernos. Close to Mall of the Emirates for shopping, cinemas and skiing (indoors in the big fridge)
Hope you find somewhere that suits you