#5 Mike,
Disagree.
Destabilization can, over time, radically alter the balance of a game.
It works best on a Large/Huge map. (If at Small/Tiny, simply blitz.)
To accomplish, start at minimal settings (i.e., the first "peg" on the slider) and wait for either: a) the I-League to be formed, or

the first report of a defection (from the targeted AI empire). You do not want to be discovered destabilizing. If you are, your diplomatic relations will tank, and the targeted AI will immediately implement countermeasures.
Once the first defection occurs, up the ante. Increase destabilization to the next spending level (slider "peg"), and offer a DOW (declaration of war - an "Attack X treaty" to another computer player) to the target's nearest neighbor (or, on "clusters", largest competitor).
Also, be careful not to neglect News Networks...Once you have declared war, the target AI will begin to attempt to return the favor and try to destabilize you.
Once the ball is rolling, you will see the enemy AI's empire start to crumble. This will accelerate as you (or your ally) is succesful on the war front. You can "save" yourself a number of turns (and transports) picking up breakaway systems on the serious cheap. I myself have taken PQ30+ systems with 1 (!) combat TR once they broke away and declared independence. (The I-League has its own tech defense rating, which is fequently inferior to both your own and the target AI's when it declares independence.)
Hope this helps.
~SDC~