In terms of "what would actually happen?" I'd vote for either the string breaking if you're sufficiently anchored to avoid being pulled along with it (since eventually the strain placed on it would exceed its tensile strength), or you getting sucked in along with it if you're not sufficiently anchored.

However, what little I know of the physics involved indicates that, from the perspective of someone outside the event horizon, the bit going into the singularity would never visibly complete the journey across the event horizon due to the extreme time-dilation effects of the singularity's gravity; it would simply go into an eternally-darkening red shift, growing fainter and fainter though always (perhaps infinitesimally) visible.