Why have minorities groups gone to the Supreme Court to outlaw the at-large system?
Minority groups might dislike this sort of voting system because it makes it more difficult for them to elect people to city councils who will (they believe) represent them. This would be true if there are areas of a city that are heavily minority while the city as a whole is not.
If there are pockets of minorities in a largely white city, minorities will want a ward-based system with geographical voting districts. If such districts exist, it is much more likely that the minority pocket will be able to elect one of its own to city council. By contrast, if all the seats on the council are voted on city-wide, the minority pockets will not be enough to elect a minority to the council.
At-large elections give more power to the majority of the city as a whole and do not allow pockets of minorities to elect one of their own. For this reason, minorities will often dislike such voting systems.