There is a basic tool called a "screw extractor". It is a hardened steel screw with a reverse thread. Your local hardware store will have it. If possible go a little larger when putting in the new screw; or put a small wood sliver into the old hold to give the new screw a good bite so it will stay firm.

I don't know ofcourse what kind of pop up spray head you have but in general the top will srew off and then you can remove all the inside parts .. leaving the base in place ... most of the time this works if the bottom is not damaged. A hint : an old serrated knife or pruning saw can be used to cut grass away from the head .. making room so you can get one or two Channel lock wrenches in to loosen the top of the head. If the base is also damaged sometimes you can unscrew it from the nipple below .. without digging a hole ( but it's a tricky move ) ... add the base only and flush it out then add back the top and nozzle. Sometime though you just have to dig the darn thing up. Good Day ...

I am with worthy, have to go pretty deep in the north. I use a footer of some kind, set the post with some dirt so it stays level, then fill with concrete. Before all of that I encase the entire part of the post going below ground to about 2" above ground with foundation coating (tar). I experimented one post with just dirt (tamped well) but was not happy with the amount of wiggle I got with it. After using concrete it was good. So people that say no concrete I dont particularly agree with that. If I can get a post to wiggle due to tamped dirt and the post is 5 feet deep in the ground, then how can you recommend dirt only. I dont get it. Maybe with clay soil you can use just dirt, but if your dirt has any sand to it dont try dirt only. As far as removing a broken post with concrete, technique depends on how deep it is. CFV's technique works fine if its not deep. For something deep you for sure will have to use a farm jack and possibly have to back fill the bottom till its out. I knew what I have have set would be a royal PITA to remove in the future, that's why I went the extra step with the tar. Make sure you get the end grain too if you do this because it sucks up water to the rest of the post like the elementary science project celery in colored water.

well, rats, I realize now, it was to late to remove, when I tried. I never saw the edit button once I wanted to remove the post. Thanks Raven I will go and remove this post in a few minutes. :) Moni