Starfox
Hero
Little of both, sometimes. I tend to start some plots running, and see which ones the players are likely to meddle in. If they ignore a given plot, it will probably affect things later on. However, I'm careful not to have a plot that nobody's interested in turn out to mean Very Bad Things if they ignore them.
This seems a good middle road for me, even if it is in some ways a "faux sandbox". It gives the players the illusion of the freedom of the sandbox, and "rewards" them by having them pick up the plotlines that turned out to be the major themes of the campaign. It is still a lot of work, tough, as you must present all the paths not taken.
I must admit part of the reason I don't like sandbox play is that I'm lazy. It is much easier for me to prepare one plotline with occasional player-driven excursions than to make a world full of plots, only a few of which will ever come into play. But this is not all - thinking up a plot hook and then not use it is hard for me. Let me give an example:
In my Savage Tide game, I very much enjoyed the first chapters, in Sasserine. I looked up possible scenarios I could use for the setting and inserted those (about half the stuff we do in my Savage Tide campaign is not actually Savage Tide). But I ended up with so much surplus material that it turned into a separate "Sasserine" campaign where the players were police. I think the same could happen if I tried to make a sandbox full of plot hooks - I would become so interested in some of the unrealized hooks that I'd want to make separate campaigns about them.