Villains usually start out as opposites to a specific hero or team, and stay that way throughout their existence. Some of the most successful and popular ones sometimes branch out to other heroes for a short while (Lex Luthor takes on the Justice League or Batman, Joker messes with Superman, The Riddler goes up against Green Arrow or Flash), but they always return to their roots and continue to be specifically seen as adversaries of "their" heroes.
However, some villains lose their roots completely, and becomes non-specific threats that can show up to hassle pretty much any hero or team. Off the top of my head we have Solomon Grundy (started off as a GL Alan Scott villain), Gentleman Ghost (Hawkman), Deathstroke and Cheshire (Teen Titans), and I'm sure there are lots more. Which ones have I missed, and what makes a villain lose his/her connection to a hero enough that he/she suddenly becomes anyone's game?