How do you get the Season 2 Reward - Silver Solo?
It was a ranked reward. Riot's description in the client reads: “Earned by achieving Silver in the 2012 ranked season in Solo/Duo.”
Is the Season 2 Reward - Silver Solo still available?
No. The Season 2 Reward - Silver Solo was a ranked reward. To earn it, you had to reach Silver in Solo/Duo during Season 2 (2012). Riot does not hand out past ranked rewards again, so you cannot earn this icon anymore.
