The best roles for solo Q, in my opinion are mid and jungle. Mid, because you're supposed to play an assassin, so you'll have a lot of damage...that results in a lot of kills and taking objectives faster. And jungle, because you have a big impact. You are the one that controls the dragon and baron...and that means a lot, especially if you know when to take those objectives.
Others will say adc is a good role for carry, but if you don't know what your support will do, it will be harder to win your lane, unless the support isn't bad.
If you're going to play a mid laner, I recomend you to take a good roamer, such as
or
. And if you're playing jungle, you can choose between the assassin junglers or the bruisers. Personally, I really like
because if you're getting behind, you can build tanky and still be useful. Also, if you're good with champions like
don't hesitate to play them. I recomended this champions because I played them and I know what they can. But you should try some champions and then choose one or two...Make sure that they won't get banned, so don't go for
/
. They aren't that easy to play and in silver they are banned.