Let's say you pick up your airbrush and try to pull back on the trigger and it's stuck. What do you do?
First off, DON'T FORCE IT. There is dried paint somewhere and if you pull it back through the needle bearing you could cause some damage to the bearing. I would bet that most of the airbrushes returned to Badger for leaky needle bearings were damaged in this way.
I came across a video by a professional airbrusher who freed a stuck needle by pulling on the rear end with a pair of vice grip pliers. It made me cringe. A couple of my Badger manuals actually say to use pliers. In my opinion, this is no way to treat a precision instrument. In the first place, if the needle is stuck it means you didn't clean well enough. A clean needle in a clean airbrush doesn't stick. Most airbrushes have a short channel between the needle seal and the paint chamber that can collect paint. Back flushing may not be enough to clean it out. If it doesn't, find something that will fit and clean it. Always wipe the needle clean, and a dab of airbrush lube couldn't hurt.
The first thing I would try is to loosen the needle lock nut. Does the trigger work OK now? If it doesn't, then something is jamming other than the needle, and the needle can be safely removed. You can then dismantle the brush and clear the jam. This condition is very unlikely.
If the trigger moves freely and the needle is still stuck, squirt a little airbrush cleaner or solvent in the paint inlet and let it soak for a few minutes. I use lacquer thinner which will cut most anything. Try twisting the needle to see if it will break free. You may have to let it soak for a while.
If the needle still doesn't want to come out easy, remove the nozzle. It could be that this is where the needle is stuck. If it is a compression held type nozzle, remove the head and pull it straight off. Be careful not to damage the needle. If it is a threaded nozzle, unscrew it.
If the needle is still stuck, try pushing it forward lightly. If the needle comes free, push it forward enough to wipe it clean with some thinner. If it doesn't come free, drop some thinner in the front of the brush, let it sit a few seconds and try again. If this doesn't work, you are probably going to have to leave the front part of the airbrush soaking for a longer period of time, or use a stronger cleaner or solvent. But, be careful of the air valve assembly. I use lacquer thinner, and the little O-ring in the valve will swell if it gets wet with thinner. I have heard that it will return to normal if left to air dry for a day, but why risk it? Also, be careful you don't damage the exposed needle.
Once the needle is free and clean, pull it out, give the brush a good cleaning, and you should be back in business.
Postscript
I came across an article once that claimed that a needle stuck with acrylics can be freed with hot water. It would be worth a try. I used to have a link to it, but the link is now dead. Thanks to John Meshkoff for pointing that out. I heard from Andrew Western that the hot water actually worked for him.