terrible movie LA Apocalypse the planet's core destabilized causing the fault lines to erupt, magma spilled out, earthquakes going every 5 minutes. Eventfully things were resolved by detonating a nuke in LA (typical American Logic - aliens on the doorstep? nuke'em. Foreign country going to launch a full scale attack? nuke'em. Planet's core isn't working? nuke it).
However assume for instance that the core de-stabilization of the core and the effects on the world was predicted and someone (government maybe?) thought it be a good idea to prepare by finding a spot which will be the least affected by the coming crisis but also be the easiest to build a new settlement (open ocean is a fine idea but you'd need to build an ocean platform first which itself has its own difficulty).
So where on Earth would this place be? and what (if any) difficulties would the people of this settlement still face with the destabilized core?