Top: Yorick, Cho'gath, Teemo, Kayle
Mid: Galio, Mordekaiser, Diana*
Jungle: Cho'gath, Lee Sin, Jarvan IV, Malphite*, Amumu*
ADC: Vayne, Caitlyn, Ezreal, Graves
Support: Taric, Leona, Nunu, Sona, Alistar*
*'s are often banned.
The reason I list the champions I listed is because all of them can carry and all of them are viable at all stages of the game... They also either have high win %'s or they are simply difficult to counter in lane.
Yorick and Kayle top, for example, counter most solo-tops.
Galio is to mid lane what Yorick is to top lane.
Vayne snowballs hard.. Caitlyn out-pokes just about everyone.. Ezreal and Graves are great bursty ADC's who can secure early kills..
Taric, Leona, Nunu, Sona, And Alistar all have great lane presence, can get their ADC's fed, and transition well into late game.