How stable is the Gulf Stream? Could it stop suddenly?
There's a pretty good rundown of this issue over at RealClimate, and a wikipedia article dedicated to the topic.

A summary of the main points from there:


The Gulf Stream is part of the Meridional Overturning Current (global ocean conveyor), which is primarily wind-driven, and secondarily thermohaline-driven. The Gulf Stream itself is also primarily wind-driven.
There has been some reduction of the thermohaline drivers in the north Atlantic, due to increased temperatures, and influxes of fresh water, largely from melting of the Siberian permafrost. This part of the system does fluctuate (as do most parts), but there does appear to be a trend.
These changes are not enough threaten the Gulf Stream, and there hasn't really been any observations of an over-all slow down that couldn't be otherwise explained by natural variation.
It is likely that the Gulf Stream will slow down over the next few centuries (slow downs appear in some of the IPCC models out to 2100).
A complete shut-down is possible (it has happened before, notably during the Younger Dryas), but it is hard to predict, as causes are not very well understood. But it seems unlikely to happen any time soon.