Well, I'm getting kind of tired of explaining this (again), but here goes.
Induced hum is basically an external electromagnetic fields (from neon, house wiring, or whatever) that's crossing the path of your pickups.
There's only one surefire absolutely guaranteed way to get rid of it, and I'll tell you what it is in a minute.
But there have been many approaches to try and mitigate it, one of which is "humbucking" pickups, where the idea is one coil cancels out the hum picked up by the other.
The guitarnuts link is excellent, and it talks about "good wiring practice" (like star grounding for instance, which is always a good thing, assuming it's done correctly).
In many cases, wiring your bass "correctly" according to star grounding and other principles of good audio wiring, will alleviate "most" of the problem, or at least make it not so bad that you can't live with it.
For some reason, guitar manufacturers don't seem to know ANYTHING about wiring pickups. Not only do they generally do a crappy job with the wiring inside the cavity, but they usually ground one side of the pickup.
This is an absolute bozo no-no if you're looking for a quiet instrument. Here's why:
In the best of all possible worlds, "ground" means the same thing everywhere. In other words, your AC outlet is grounded, your amp is grounded, and your guitar cavity is grounded. In a perfect world, "ground" would mean the same thing in all those three places.
But that's almost never true in the real world. For instance, you'd think that if your amp is grounded, then your guitar would be grounded too, because of the shield on the instrument cable that connects them together.
But the shield isn't a "perfect conductor" like a wire, in real life it has resistance, inductance, and capacitance, and actually it's more like a "transmission line" (in the radio and TV sense) than a "wire".
Therefore, and depending on the circuitry inside the amp, there may be a difference in electrical potential between the "ground" that's in your guitar, and the ground that's on your amp chassis. This potential difference creates something called a "ground loop", and it effectively turns your whole pickup-cable-amp system into a huge antenna.
That's why, you can sometimes make the hum go away (or get worse) by touching the strings on your instrument. What you're experiencing there is the effect of body capacitance on the transmission line.
That's also why, you can usually get better results with a short instrument cable than a long one (try it, use a 3 foot cable and see if some of the hum doesn't go away).
Now, the problem arises when one side of the pickup is "grounded" in the instrument cavity. If the cavity "ground" isn't the same ground that the amp sees, the return path from the pickup to the amp ground will include the long instrument cable!
The only surefire way to alleviate this problem is to make sure that NEITHER SIDE OF THE PICKUP IS GROUNDED!!!
The only way to do this is to use a balanced connection between your guitar and your amp (or equivalently, and probably cheaper and easier, use an onboard differential input active preamp, which will accept a balanced input from your pickups, and convert it into a low impedance single ended output suitable for driving long cables).
Crusty old audio engineers have been advocating this method for years, but for some reason the guitar manufacturers haven't found it cost effective to do this. And in fact, balanced lines are the industry standard for microphones and many other low signal-level audio devices.
If you want a schematic diagram for an inexpensive differential preamp that you can build yourself for around 5 bucks, drop me a PM. You'll have to put a battery in your cavity though, 'cause the preamp will require a power source. It won't change your tone "at all" if you can match the input impedance to your pickups (which is very easy, it just involves selecting the right value resistors).
But I guarantee that this will alleviate hum problems in 99.99% of the cases. There are some exceptions (for instance there are some funky pickup designs, and some funky amp designs, but by and large for "normal" instruments and "normal" amps this method will work perfectly).
As a case in point, the very first F bass I ever bought (about 10 years ago) had a horrible hum problem when I first got it. I could move around the room and pick up all kinds of interesting stuff (hum, staticky noise, radio stations, etc etc). So, I installed a differential preamp for each pickup, and that completely cured the problem. Now, I can take that bass and plug it into my loudest 2100 watt amp, turn all the volume controls up all the way, and stick the pickups right next to the ballast on a fluorescent light fixture (not more than 3 inches away from the light bulbs), and it's quiet as a church mouse. Nary a peep. It's quite amazing really.
