It never fails -- if someone complains about noise in any part of the sound system, they will describe it for you as "static."
Sometimes you can get them to imitate the noise. Tom and Ray, the Car Talk brothers, demonstrate this every weekend on their radio show. But most times, what you have to do is put your own ears in the space and hear it for yourself.
The most frequent sustained noise you get when mic'ing a pit or in monitor speakers is hum -- 60 cycle for us in the States. Once you've heard it, you always recognize it. Sometimes it can be tracked to, and corrected with, proper grounding. Sometimes people try to correct it by defeating the safety grounds -- a practice I do not recommend.
(Also, because defeating the safety ground often stops the ground loop of the moment but leaves the device unprotected for the next surge of electrical noise in the system as the lighting design reaches a point where the dimmers are making maximum noise in the building wiring).
The other instantly recognizable sound involving microphones and speakers is feedback -- which the British refer to by the wonderfully descriptive phrase "Howl-around." When you have been running mics near feedback for a while, you will also recognize the distinctive "ring" of near-feedback conditions. Basically, feedback is a loop between speaker and mic that is higher than unity. The ringing is when the loop is under unity; the tone of the feedback of the moment starts, but can not sustain.
Then we get down to crackling and popping. Soft crackles are usually dirty connections (why I always travel with a can of Caig De-oxit) or dirty pots. Extremely loud and sharp pops come when there is a bad connection or a broken cable to a microphone with phantom power on it. And a single loud "pop," not repeated, is due to the bass player yanking the cord out of the DI three seconds after his last note so he can get packed and off to his next gig.
As a general note, electrical connection problems have a crackling sound with a lot of high-frequency content -- this is how you recognize them as being something other than rattling drum hardware or a failing microphone.
Hiss comes from a great many sources. There is always nearly-pure white noise in electronics; the basic thermal noise of the components. If any stage is gained too high, you may pull that noise floor up to where it gets objectionable. The only real recourse is to gain-stage carefully -- putting stronger signals down the lines from the earlier stages so the later stages don't have to amplify the noise floor along with the wanted signal.
Especially when you try recording in an ordinary stage space, you notice another hiss; the noise floor of the building itself (ventilation, air movement, traffic noise from outside, etc.) This is more of a low-mid ranged whooshing sound, plus of course any louder specific elements (like a noisy fan in the HVAC). One place where this may become objectionable and get mentioned to you as "I heard this hiss during Laura's lines" is when you are compressing or gating the microphone in question. Aggressive settings of either will mean the noise floor can be heard coming and going (referred to as "breathing" or "pumping.") A steady, constant noise is less noticeable than one that changes quickly.
With powered monitors, especially, there is another extremely familiar noise you will hear; GSM noise. Which always means that someone put their purse -- with their cell phone in it -- down on top of a monitor or on the floor right by some insufficiently shielded electronics. I suspect an entirely new range of ugly noises of this type as the even more powerful radios of the latest round of smart phones become more ubiquitous.
Wireless of course has a whole slew of new and wonderful noises.
There is a constant reedy whine of specific frequency, often sounding much like the back of an old television set. This is a fault in the transmitter itself and means it needs to be returned to the company that made it. It seems to happen as certain brands age.
There is an extremely loud broad-band white noise. This is usually due to interference, specifically, one of the new DTV stations. There is no space here to go into the Digital Dividend, the restructuring of the airwaves, and so forth, except to note that even when you are in the white space (as you should be!) under the right conditions your receivers may still reach for a strong but distant television signal instead of the nearer (but extremely weak) belt pack you wanted.
Related to this is when there is a momentary interruption of the wireless signal -- due to path cancellation usually. Since the wavelength is in centimeters, the physical location in the building where the multiple transmission paths meet out of phase and cancel is a mere centimeter or two in size. So with an ordinary actor, doing ordinary acting, all you will hear is this distinctive "Fwiip!" of white noise as the receiver loses lock for a moment then resumes. You can usually "fix" this by turning the squelch higher -- at the cost of having the signal cut out completely for these brief moments.
When the drop-out is complete; aka, the receiver properly squelched, but it is a significant time before the signal is regained, you can often hear this as a dull popping sound. This is something -- like many RF issues -- you need eyes as well as ears to diagnose.
Sometimes a constant source of interference can hetrodyne with the transmitter in such a way that what you get is a low-level modulation; it is a sort of siren whistle, usually low in level, that goes on almost constantly; "weewooweewooweewooweewoo...."
On the wireless elements themselves, the most frequent failure you will hear is a broken connector. This is the element that gets the most stress, and the place where it fails most often. The connection itself can come adrift; the Sennheiser connectors have a nasty habit of unscrewing, the Shure connectors over time will lose the spring in the detent button until it no longer locks the connector. For both, electrician's tape is often the stop-gap to keep the connector secure.
The wiring also rips, either just inside the connector or just above the strain relief. It can also be cut higher up, especially when someone has been careless with bobby pins or toupee clips. But the tear is usually within a few CM of the connector. These will show up as a loud crackle, and can usually be confirmed by manipulating the cable.
Sometimes you have a intermittent and usually softer crackle due to corrosion of the contacts in the connector. This is, again, why Caig De-Oxit is in my gig bag. I tend to spray the connectors every time I put a fresh element on.
When the sound is hollow or sounds like it is flanging, the first place to look is if the element has slipped, or if the actor has managed to put the tape on top of it. This is the same sound you will hear from the actress who thinks that holding her mic to her face with one hand is sufficient for mic check (she will attach it later after she has finished with her make-up.) It is also (although in a different frequency range) the lovely sound you will get used to on any period show where half the cast are wearing hats.
Sweat makes a whole variety of sounds. Sweat is sufficiently conductive so it can short out a mic if it gets far enough. You will hear this as loud but dull-sounding pops most of the time, followed by complete mic failure, and then followed by a mysterious resurrection.
It can also block the screen or sit on top of the diaphragm and make a clouded, distant sound. I had a crop of Senny ME2's that got sweated out, and they produced this very interesting illusion that the actor was ten feet away and was singing as if his heart wasn't really in it. Countrymen will get quiet, shut off for a while, restore themselves...but still sounding clouded and muffled.
My two major activities against sweat these days is to wrap moleskin around the cord just behind the element (it traps much of the sweat that would otherwise drip down the cord) and to rinse out the mics in isopropyl alcohol several times a run (and let them dry THOROUGHLY before powering them up again).
Over their lifetime, Countryman B3's get dull-sounding, subtly but increasingly muffled until you have to retire them. The cleaning in alcohol will often help. Shure WL93s, which I was graced with the opportunity of working with at far too many venues, fail more suddenly and more thoroughly; the diaphragm becomes detached. Suddenly the output drops dramatically and seems to come and go. Sometimes, the detached piece is exactly right, though, leading to this rather lovely sound of tiny chimes that sound along with the voice.