Something here seems obvious to everyone but me...

Ok, Say I declare a goal... "Rob the bank", or "Convince Lois I am not a world famous superhero."
Other people could claim the opposing side.

But supposing no one cared? Perhaps there are other more interesting conflicts to claim...
what happens? Do I automatically win? Or does it just hang there waiting patiently till someone claims it before being resolved?

Hangs around waiting.  The scene doesn't end until all conflicts are resolved, and the only way for a conflict to resolve is for someone to claim a side, and then for that side to win. 

The fact that you played the conflict has absolutely no bearing on who ends up claiming and winning it.  You do NOT have an automatic claim on it.  You have to claim at the beginning of the page just like everyone else, and if you are late in the turn order both sides might be claimed by the time it gets around to you. 

What Hans said. If you created a Conflict and nobody wants to claim it, including you, it stays there until someone does. If the next round you claim it and roll on it and nobody opposes you, then yeah, you're pretty much going to automatically win it.