How do I remove plastic shelf brackets that have broken off in the hole?
Plastic plug in a hole, I've always found drilling into it with a bit slightly smaller and stopping the drill, then pulling it straight out works dandy. If not, it gives to a place to try using a screw (and a nailpuller on the screw if it's REALLY stubborn) more easily. Then look for metal replacement brackets in the correct size for your bookshelf so it does not happen again.