The corkscrew is in the other house, or the drawer, or it was never in the picnic basket at all. Half the advice on this problem is theater, and one or two of the popular methods are a genuine way to end an evening in an emergency room.
First, check what you are holding
None of this applies to sparkling wine. A sealed sparkling bottle holds several atmospheres of pressure, and its cork comes out with a wire cage and thumb pressure, never with force, heat or an implement. If the cage is missing, hold the cork, twist the bottle, and keep it pointed away from everyone.
For still wine, look at the closure before you start. A screwcap needs a firm grip and nothing else. A synthetic cork is tougher and grips harder than natural cork, which matters for the methods below.
Bottle: still or sparkling, and never force a sparkling bottle: Closure: natural cork / synthetic / screwcap: Method chosen: Surface: solid table, nothing that can crack: Cloth over the neck: Sieve or coffee filter ready:
The method that actually works: a screw and a claw
Drive a long screw, ideally a wood screw two thirds the length of the cork, straight into the center of the cork, leaving a couple of centimeters proud. Hook the claw of a hammer under the head and lever it out slowly and straight. A screwdriver and pliers do the same job with more effort.
This is the only improvised method that behaves like a corkscrew, because it is one. Keep the screw central and pull without twisting the bottle.
Pushing the cork in
Stand the bottle on a solid surface, put a cloth over the neck, and press the cork down with the blunt end of a wooden spoon using steady pressure rather than a blow. Expect a spray of wine, which is what the cloth is for.
The cork then floats in the neck and interferes with pouring, so decant the wine into a jug through a sieve. This method is reliable, undignified, and perfectly fine for a young everyday bottle. It is a poor choice for anything old, where the cork is likely to break into fragments.
The shoe method, honestly
It works. Put the bottle base-first into a firm-soled shoe, hold it horizontally against a solid wall, and tap the sole against the wall repeatedly. The cork creeps out a few millimeters at a time over ten or twenty taps, and you finish by hand once enough is showing.
The risks are real: a chipped base, a cracked bottle, a hand full of glass. Do it against a wall rather than a floor, stop while the cork is still gripping, and never do it with a bottle you would be sad to lose. It is a party trick with a failure mode.
What not to do
Do not heat the neck of the bottle to push the cork out with expanding air. Glass under uneven heat breaks, and you are holding it. Do not strike the bottle against a hard surface. Do not push a knife blade into a cork while pulling toward your own hand. Do not saber a still wine bottle indoors, or at all, unless you know exactly what you are doing.
And do not use force on a bottle you are gripping by the neck. The combination that hurts people is always the same one: pressure, glass, and a hand wrapped around the thing that is about to break.
All of these are improvised methods and all of them can break a bottle. If the wine can wait, waiting is the safe answer, and a basic corkscrew costs less than a hand injury. Anything below is at your own risk.
Crumbling corks are telling you something
If the cork disintegrates on an older bottle, push the remainder in and strain the wine. Then note it. A cork that fell apart, a wine that seeped past it, or a fill level lower than the age suggests are all clues about how the bottle was stored, which the fault guide covers reading before you blame the wine.
Record the opening either way, with whatever you learned about the bottle. The opening history guide covers keeping that note attached to the wine, and if the bottle is not finished, the leftover guide covers what happens next.