This sounds like a shielding problem. There's a source of RF near/in your practice space, and you're picking it up. Single coil pickups?
It could be a shielding problem, but when I hear buzz I normally think "dirty hum"; 60Hz hum coming from a "dirty" power source with micro-spikes in the voltage. RF "white noise" is generally more of a hiss.
If the buzz is present in one room and not another, suspect #1 is the wiring of that room. First of all, make absolutely sure that every plug in that room is grounded. The ground pin is generally connected not only to fault circuitry in the amp but to the metal shell in the amp that provides at least some shielding of the amp circuitry from the transformer and ambirent noise. If the outlet is ungrounded, so is that shield, and EMI will go right through.
Also, wiring in many buildings is done using metal wiring conduit, which makes it easier to feed wires down into outlet boxes and protects the wires from damage which would otherwise result in an electrical fire. That conduit is grounded and so drastically reduces 60-cycle hum in electronics. If the room was built with "naked wires" not enclosed in metal conduit, you are surrounded by hum-broadcasting antennae. Re-wiring the room is probably not an option (though not using wiring conduit is a fire hazard and thus a violation of building codes in most jurisdictions, so it will eventually have to be redone)
Suspect #2 is the presence of high-voltage wires near the building. The building's wiring could be up to code, but if there are high-tension power lines or even an inordinate number of street-level poles within 50 feet of your practice space, the hum generated would be substantial.
Because this is happening only to your rig, I would check the grounding of the outlet you plug into and then the outlet others plug into. Try swapping outlets, and if others have a problem with that outlet while your problem goes away then avoid using that outlet.
The next cheapest thing to check is that your instrument cable is adequately shielded. Borrow the cable of someone not having a buzz problem and see if the buzz goes away. If it has, ditch your cable and buy a new one. Get a cable only as long as is needed; the longer a cable is, the more you will lose signal and introduce hum. Consider a wireless system if you need a cable longer than 15 feet to do what you normally do on stage. And make sure it is an instrument cable and not a speaker cable. Speaker cables are unshielded because the current passing through them is low-resistance, high-voltage and thus not susceptible to EMI. The front end of the amp circuit on the other hand is exactly the opposite.
If you have more than one bass, try swapping basses. If you have no other instrument, try plugging a friend's guitar into the amp, preferably one that isn't having a buzz problem (don't worry; bass amps can handle a guitar signal, unlike vice versa). If the hum is lessened with one instrument versus another, the problem is your bass; you'll need to invest in humbucking pickups and shield the control and pickup cavities to reduce the noise. Most basses have either humbuckers or neck and bridge single coils that hum-cancel, but if you have a P/J, a five-string with split-coil HBs, a '51 P, or are blending toward one pickup or the other in a single-coil set, the hum-cancelling will probably not be 100% effective and you should consider an upgrade.
Also try repositioning the amp and see if the noise is lessened. If so, find the angle at which the noise is at a minimum and keep it there. Do the same with your bass; if there's an angle you can stand at which noise is at a minimum, stand there. Bass isn't as directional as treble frequencies, so which way your cab faces doesn't make as much difference as it does for the guitarists.
You can invest in a noise suppressor; this will filter out signal at 60Hz and harmonics to remove hum (invaluable for single-coil guitars and basses with soloed SC pickups). However, by doing this you are usually also reducing or removing those frequencies from your bass' sound. Most noise suppressors are tone-suckers, and good ones are expensive.
If that fails to yield a solution, the problem is your amp. The cheapest way to reduce or eliminate the hum in the general case would be to first make sure your amp circuitry is completely shielded. Usually if your amp has an aluminum shell on all sides this is not a problem, but you may find that there is no shielding between the transformer and the rest of the circuit, or that the top or bottom of the amp has no metal between the circuits and the outside world. Find rolls of fine copper mesh; This stuff isn't normally found in your everyday home improvement or electronics store, but if you know of a nearby tinker-tronics type store that caters to electronics hobbyists, they would be your best bet. Simply secure this mesh around the transformer, especially between it and the amp circuitry. Take care that the mesh does not touch any amp circuit or component other than the shell (the transformer itself is generally safe to touch, but doing so might defeat the shielding or reduce power. Anywhere else the copper touches will probably short-circuit the amp). Also ensure that any side of the amp circuitry NOT covered by a metal side of the amp shell has a layer of copper mesh over it (it would probably be easier to spread the mesh over the inside of the non-metal cover on that side rather than span it across the top of the circuitry). Make sure the mesh has a good path to ground, either with good contact with the shell or by tack-soldering a wire between the mesh and ground.
As a last resort, you can protect your amp and bass by turning the practice space into a Farraday cage. Using a LOT more of that mesh, paper the walls, floor and ceiling with the stuff. Make sure the mesh is electrically continuous around the room, and then connect that mesh to the ground of an outlet in the room (please be EXTREMELY careful when tinkering with an outlet and make sure you cut the circuit breaker to that outlet before working with it). Generally the screws that hold the cover plate on are shielded; solder a length of wire to the shield and wrap the other end around that screw and you're good to go. Obviously this is expensive to do right (as in hiding the copper mesh), and as it doesn't address any potential instrument, amp, cable or house wiring problems, it's really only a good solution if the problem is a high-voltage line just outside and there's nowhere else to practice. The Farraday cage will also prevent useful radio signals from entering or leaving the room. You will have to leave the room to use any device that communicates via radio frequencies, such as mobile phones, AM/FM/XM/CB radios, Wi-Fi or cellular Internet access, etc.
