is the cork sticking out of the bottle at all? or is the cork broken, or waterlogged at all? ive gotten mine out by just using a pencil or something of simaler shape to poke the cork and break it, and then just turn the bottle upside down to get the peices out. or try an actual cork remover like you'd use to get the cork off of a wine bottle.
i use a long shaft screwdriver. set the bottle on something firm like the kitchen table. push the screwdriver into the cork and break it up. gentle pushes so as not to break the glass!!
take a piece of a coat hanger and bend a U at one end, half the size of the mouth of the bottle. turn the bottle upside down and hook it or get behind it and pull it out. some soapy water will help prior to pulling. I've done this for many years and never broke one.
If it's stuck in the opening, I use a regular old corkscrew....probably doesn't HELP any value at all, but shucks, I'm not looking to sell my bottles. I just put the corkscrew in far enough to grab the non-rotten part of the cork, and be very gentle.
all great ideas but heres one i use, i use a drywall screw to screw into it and then pull it out. if it's inside the bottle then i leave it in or put my brush in the bottle to the base, turn it uside down and make sure the cork is at the bigining of the neck, and pull it through when pulling the brush out, not to be done on thin bottles.
Years ago I use to have a cork remover that had a long pin with a hole. Push it through the cork. Pump the handle a few times and pop goes the cork.
It left almost no mark.
I have been looking for one, but cant find anything like it.
Please keep in mind, that if you are opening a bottle that contains chemicals this could produce a lot of dust.