A shower arm screws into a brass 90-degree fitting behind the wall, and all you should have to do to remove it is grab the arm and give it a good counterclockwise twist. The only help you should need to do the job are a rubber glove and, possibly, a pipe wrench (although a strap wrench is a better alternative). These won't help, though, if the shower arm is corroded and about to break or has broken already. It could turn out that you didn't have a leaky showerhead but a leaky shower arm. Don't worry, because getting it out will probably be easier than you think.
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Normal Removal Procedure
If your shower arm was installed within the last five years, you should be able to remove it without tools. You may or may not deem it necessary to turn off the water to the bathroom, but you should definitely keep the shower faucet off while you do this. If the shower comes on while the shower arm is off, water will get behind the wall and potentially cause mold problems.
If you have a bent shower arm, the best place to grab and turn it is right on the very end. You don't have to unscrew and remove the showerhead first — that's up to you. If the shower arm won't budge or it isn't bent in a way that gives you a mechanical advantage, wrap a rubber glove around the pipe or put it on to get a better grip. The vast majority of the time, the arm will come off.
The Shower Arm Is Stuck
If the shower arm has been there for a long time, the threads may be locked by rust and scale, and where there is rust, there is often weakened metal that can break if you rely on muscle alone. You need to get a look at the place where the shower arm connects to the plumbing, which means prying off the escutcheon with a flat-head screwdriver so you can peer through the hole in the wall with a flashlight.
If you see white deposits around the connection, that's caused by minerals, and without access, there's not much you can do about it but keep trying to turn the shower arm. In the absence of scale, a few squirts with spray lubricant may loosen the threads.
It Broke! Now What?
No matter how much you try to avoid it, a corroded shower arm that's stuck is bound to break if you use too much force to try to turn it. This may seem like a serious problem that will force you to break through the wall to replace the fitting with the broken pipe stuck inside it, but there's a simple trick that can help you avoid all that. All you need is a standard 10-inch metal file, a hammer and your wrench.
Poke the tapered end of the file through the hole in the wall, insert it into the fitting a far as it will go and tap sharply with the hammer to wedge it against the sides of the stuck, broken pipe. Clamp a wrench onto the file, then tap the wrench handle sharply with the hammer to break the seal locking the threads. If the file spins inside the pipe, tap it harder with the hammer until it doesn't.
Remember that the pipe has to turn in the counterclockwise direction to come loose, so make sure you're tapping in the right direction. A few good taps should break the seal so you can finish the job of unscrewing the pipe and extracting it by hand.