Some maths would have to be worked to see what cases did not just shoot off into space.
An elevator whose centre of mass is in GS-orbit would happily sit in that[1] if sheared off
at the base, but if that was sheared off any distance up, the CoM would now be in higher orbit, and that (simplifying things) would be travelling faster than the natural orbital speed at that point. (It's possible that it would have an elliptical orbit whose perigee was at that height, but swinging further out for the rest of the time.)
But most SE-concepts have the counterweight giving significant (FCVO 'significant') outwards pull, such that the SE is in constant tension.
Any cut of the cable (from ground level upwards) gives an outward force. The maths says that there could be stable orbits, too, but the higher the cut, the more likely it is that the initial tangential velocity gives it escape velocity.
And in the specific case of a multiple-point failure (including deliberate use of demolition charges to create smaller fragments, when an initial severing is created), I'm pretty sure (from maths done by me a
long time ago, so open to being wrong, here...) that a counterweight station disconnected from the GS one will fly off out of Earth's orbit. That would be true unless it was a truly massive counterweight close to the 'mid-point' at GS, IIRC. Actually, given that the use of the counterweight for launching interplanetary probes (I always envisage a floor hatch, like that of a bomber, and people pulling a lever at the appointed time to just 'drop' them into space) without any additional launching fuel, I think that we're always going to be pressurised (scientifically/economically) to deliberately put the counterweight beyond that critical point, even if we needn't.
BTW, I forgot to say to someone's previous half-question, that space elevators based at non-equatorial latitudes
could be stable, only increasingly cumbersome and angular from the vertical at the point they touched the ground.
(Extreme case, to not intentionally refer to another poster's question: very near the pole[2] you might well be able to have a space-elevator travelling effectively horizontal from the anchor-point until it 'carries straight on', where the ground falls away, and heads up towards some sort of GS-like orbital point where a station is actually pulled into a "higher latitude" GS-orbit than a free object would be, and then onwards to the counterweight (again, slightly off of the ecliptic, but less so). You'd need (roughly, give or take the difference due by Pythagoras, and of the catenary-like 'hang', towards the anchor end) cable of a length from Earth's centre to GS-orbit, however, rather than just cable of a length from Earth's equator to GS-orbit. On the other hand, if you were happy with a polar-located anchor end being pulled in all directions, not just one, you could set your structure spinning at a speed other than Earth's spin rate, for longer or shorter structures. Imagine one engineered (with tolerances to account for the perigee/apogee and off-elliptic directional differences, of course) to track the moon! Good luck with engineering
that one, though...
)
Noting that while it would be potentially a stable equilibrium, in the perfect case, there'd be those undamped oscillations already mentioned. For better or worse. It might still be like working at the end of a very long, slow-period pendulum. It
wouldn't be like standing on top of a pile of milk-crates, where going too far over[3] is easy and leads to the inevitable fall.
I find the Space Fountain concept interesting, but the fact is that a loss of power results in the tower that you have no longer being 'held up' by the stream of whatever-it-is-you're-fountaining and only sitting there while it's unstable equilibrium point is still being stuck fairly close to. The 100km-plus 'vacuum tube' (present in most ideas of this sort) is probably more destructive on its way down than those fragments of elevator-ribbon would be. As far as I can see. The advantage being that you've managed to place it more or less anywhere you want on Earth's surface, so can keep it 100+km away from anywhere that matters.
The speculation is interesting, though. (And, I still admit, I'm a bit out of date on the whole technology. Rusty on some of the bits I did once know about, too.)
[1] And, depending on other factors, continue to rotate in "absolute space" at the same speed as the Earth, if it had been perched up there enough to have settled down enough in that regard... prevailing winds at the base might spoil that, however)
[2] Not at, both because the catenery of the cable would dive below the horizontal at that point, unless you could build a large enough tower to avoid that, and the fact that you're actually aiming at a point slightly below the tangential horizon.
[3] Actually, you could get it going too far over, but that'd take a large amount of effort, in bringing the counterweight past a certain angle. By that time, you'll have had a lot of warning of what was happening and doubtless send out your Action Heroes Of The Hour to confront the Evil Overlord trying to slowly rain on everyone's parade. It'd have been easier for him to just hack into the safeguard demolition charge controller/anti-ballistic-threat-missile-system and blow the thing up with its own defences.