There are two types of axles I'll discuss here: rear wheel drive axles (RWD) and front wheel drive axles (FWD). We'll start with RWD. I don't often hear noises from the axles themselves, but I do hear noises from the U joints and center bearings if equipped. If you have a bad U joint, you'll often hear a loud click or clunk when you engage drive or reverse. It might be worse in one direction than the other.
To confirm you have a bad U joint, simply grab the axle and twist it back and forth and move it up and down while you feel for looseness. Bad U joints are often obvious and will move around a bit if they're bad. Sometimes they won't be so obvious; in those cases, look for orange dust around the U joint caps. This indicates a lack of lubrication and often a loose part.
If you find a loose U joint, replace it. You really can't make them better any other way. I recommend you install U joints with serviceable fittings so that you can lubricate them in the future. This will ensure that the U joints last a good long time. It's usually a lack of lubrication that kills them in the first place. One last tip: Install the grease fitting toward the axle; this will make it easier to service later.
As for center bearings, this can be tricky to track down. These are bearings that are used with particularly long drive shafts where they're split into two pieces. They use a center bearing to support the drive shaft over this long distance.
I'll admit it's difficult to diagnose a noise from this area, mostly because they often make noise under load, which means you can jack the vehicle up and run it and never hear the noise. Drive down the road, however, and the noise is there.
The best advice I can give you here is to try to get your hands on a chassis ear. This is a tool that uses tiny microphones that you place on different parts of the vehicle to help you find noises.
These tools aren't cheap, and I don't believe you would be able to rent one from your local auto parts store. They're mostly found in repair shops and are used to track down difficult noises. That said, if you have or suspect a bad center bearing, replace it. Once again, these parts aren't serviceable, so you'll need to replace them if you find a bad one. You may find that you need to replace the entire axle assembly. If this is the case you might consider finding a salvage unit. If you do it may save you quite a bit of money.