The main issue in the setting of an orchestra or choir is the fact that no two voice or instruments maintain exactly the same pitch for any length of time. If you have two pure sine wave source that differ by just one Hertz, then the interference pattern between them will shift over time - in fact at any given point you will hear a cycle of constructive and destructive interference which we recognize as beats, but the exact time when each member of the audience will hear the greatest or least intensity will vary with their position.
Next let's look at the angular distribution of signal. If two tenors are singing a D3 of 147 Hz (near the bottom of their range) the wavelength of the sound is 2 m: if they stand closer together than 1 m there will be no opportunity to create a 180 degree phase shift anywhere. If they sing near the top of their range, the pitch is closer to 600 Hz and the wavelength 0.5 m. But whatever interference pattern they generate, a tiny shift in frequency would be sufficient to move the pattern - so no stationary observer would experience a "silent" interference - even of the fundamental frequency.
Enter vibrato: most singers and instruments deliberately modulate their frequency slightly - this makes the note sound more appealing and allows them to make micro corrections to the pitch. It also makes the voice stand out more against a background of instruments and tends to allow it to project better (louder for less effort on the part of the singer). This is used by soloists but more rarely by good choirs - because in the choir you want to blend voices, not have them stand out.
At any rate, the general concept here is incoherence: the different source of sound in a choir or orchestra are incoherent, meaning that they do not maintain a fixed phase relationship over time. And this means they do not produce a stationary interference pattern.
A side effect of interference is seen in the volume of a choir: if you add the amplitudes of two sound sources that are perfectly in phase, your amplitude doubles and the energy / intensity quadruples. A 32 man choir would be over 1000 times louder than a solo voice - and this would be achieved in part because the voices could only be heard "right in front" of the choir (perfectly coherent voices would act like a phased array). But since the voice are incoherent, there is no focusing, no amplification, and they can be heard everywhere.
Note that incoherence is a function of phase and frequency - every note is a mix of frequencies, and although a steady note will in principle contain just a fundamental and its harmonics, their exact relationship is very complicated. Even if you took a single singer's voice, and put it into two speakers with a delay line feeding one of the speakers, I believe you would still not find interference because of the fluctuations in pitch over even a short time. Instead, your ear would perceive this as two people singing.
And finally - because a voice (or an instrument) is such a complex mix of frequencies, there is in general no geometric arrangement of sources and receiver in which all frequencies would interfere destructively at the same time. And the ear is such a complex instrument that it will actually "synthesize" missing components in a perceived note - leading to the strange phenomenon where for certain instruments, the perceived pitch corresponds to a frequency that is not present - as is the case with a bell, for example.
Humans hear the correct perceptive signal for a sound wave of that frequency.
We really can't say much more than that. The psychology of acoustics are very complicated and could fill volumes.
It's closer to say we have cells which act resonant at a specific frequency. Our brain identifies which cells are resonating at any point in time, and constructs the signal from that. Our brains receive information that cell A or cell B is signalling. The association between those neural signals and frequencies is a learned response that we pick up early on, as an infant or perhaps even in the womb.
Best Answer
This is a neat question. Did you know that adding two Sine waves of the same frequency but different phase together always produces another Sine wave? Of course you can imagine two perfectly out-of-phase Sine waves that "cancel" by adding to a line but in that case you can just imagine the result as a Sine wave with 0 amplitude.
Using gnuplot with the following commands I plotted the sum of 4 Sine waves:
And on the first go I got this:
Notice the shape and frequency is the same but the amplitude has increased. If you add them randomly together again you'd get something like:
Notice that the amplitude is still greater than 1 but the phase has shifted.
What's happening is that each Sine wave you add increases the maximum possible amplitude so the more you add the better the chance of getting a wave with amplitude greater than 1.
Chris White posted a fantastic answer about the brightness of the sun where he goes into a lot more details about the math of adding waves and the statistics of it.