Shield stability and poise are two different stats and are not related in terms of blocking. I'm not too sure whether there is an actual breakpoint besides running out of stamina.
For a shield to withstand a heavy weapon you want to upgrade it to 60+ stability at least. The super-light Heater Shield has 68 stability when upgraded to +14 and works pretty well for me in PvP, usually withstanding two heavy hits. The commonly used Grass Crest Shield (63 stability at +14) usually only withstands one hit.
If you want to effectively block heavy PvE enemies/bosses you're probably looking for 70-80 stability. Black Knight shields work great if you have at least 16 strength to wield it and upgrade it to +5 (73 stability). As for greatshields, they can be really useful in some bossfights (blocking multiple swipes of Seath's tail for example). I like the Greatshield of Artorias for strength builds, it's not too heavy and has a whopping 88 stability. Although you do need 34 strength to wield it...