HOW bad is it to fall on personal anchors? It's obvious-ish that it's not a good idea to fall on a short rope since there isn't much there to absorb your fall, especially if it is really static. We drop tested cow's tails, which are personal anchor systems (PAS) that cavers use.
We were hoping the dynamic ropes would absorb enough to not hurt but damn, since we were dropping 300lbs/136kg it still got to forces that were pretty gnar. Dropping the same cow's tail after the knot cinched up took it from 6kn which is survivable to 10+kn which isn't.
Just extending the length of it a bit got it to enough force to break... a dynamic rope! When you fall twice as far as you have rope in the system, that is a fall factor 2 (FF2) and that is when things start breaking.
Static material is even worse. Semi static got us above 10kn with fresh knots but then it was just breaking stuff after that. The super static polyester rope broke on the first drop. The Petzl Speleoglyca dyneema PAS didn't break on the first drop only because the stitching broke.
We cross loaded a carabiner in our drop tests. See this video to watch us do it on purpose!