I took out the rotten teeth that had been fence posts set in concrete by digging round all twenty or so of the concrete stumps , once I'd got deep enough levered the stump up to one side using "PUDLOCK's " , (scaffold poles with a flat on one end " and back filling each hole by continually rolling the block back and forth with the levers to let me do the back filling .
I have also used a half width garden trowel fitted to a broom shaft , to chip away hard packed clay and use an industrial vacuum cleaner to take it out the hole to get awkward posts out between Leylandi & rose bushes etc. all done like this so I could then take out the sodding Leylandi by the same method so Ididn't wreck next doors garden & well trimmed low box hedge
Once out the ground the concrete plug is broken fairly easily with a sledge by cracking it at the corners of the former post ...so you are walloping the thinnest & weakest part . In later life I gave up using the sledge due to my back & shoulder injuries and so used an 650 watt SDS hammer drill with a flat chisel ..set to just hammer .
I've also often used my power washer to blast , soak ,soften the soil around posts & blocks to loosen it . By the time you're at post number five , number one is usually soft enough to dig out several inches fairly easily .
Back filling holes with wet soil & a hose pipe is a good way of knowing you are going to be able to dig in a new post hole that is free of rubble & bricks in a few days time if you have the luxury of time on your side.
Putting the new posts in a wet hole using a dry mix is far easier than having to cart a wet mix , the extra wet soil soon sees it " Go off " in a few days ..even less if you've added a quick set powder to the mixing .
I have also used 10 inch long hardened steel coarse thread wood bolts with hexagonal heads and thick washers ( screw fix on line ) to fit a loop of decent strong chain ( screw fix again ) to the top of posts in concrete and use a 4 mtr 4x 4 lever over a block of apple tree bole to jack the whole base out after giving each " peg " a power washing every night for a few minutes to get the soil damp low down if its been dry .
You normally find that there is solid wood three or four inches below the top of the concrete .
There is an easier alternative sometimes available to you and that is to dig out the first post entirely back fill with very wet soil , then dig the other post holes to the side of the others in the run .
Here at this property I cut a full fence panel in half & re-made it as a proper looking half panel , so I could take advantage of the virgin ground in between the original fence line . This saved me having to dig out 21 rotted posts that had been set in concrete.
I've replaced all the old fence support poles with concreted in " H " section reinforced concrete posts and slid in new fencing panels .
Nearly 20 years ago when making a small boundary fence about 20 inches high I capped all cut sloping tops of the short fence posts with 150 mm self adhesive imitation lead flashing tape . They were still all there a year ago .
One of the biggest problems with wooden posts is that folk don't protect the base of the posts at the " Dew point " , where the effects of Dew & damp are to be found ....this is the first three inches below the ground and the first nine inches above it .
Whilst in British Telecom com it was quite normal to find almost 100 yr old telegraph poles in towns & cities as well as the countryside .
This feat is easily achieved quite cheaply , because every seven years the base of each pole was dug out to six inches down and then the whole bottom painted in creosote up to the 18 inches point , the dug out base was then back filled and tamped down .
The old fashioned mahogany railway sleeper farm gate posts are a bit more difficult as most were 9 or maybe 10 feet long ( ? ) & set in a four foot deep hole that was four feet round at the top and three feet round at the bottom to which Punnered " or rammed down rubble and sometimes a very liquid concrete was added whilst it was being set up.
That's when dad and the other farm workers dug a real big hole and used a pair of strong friendly shires to drag out the stumps on a chained pulling harness before back filling the crater and punnering in yellow or blue clay , then digging a nice new gate post hole .