Why do we hear frequencies in the basis of sine waves? When we talk about hearing frequencies and overtones, we almost always mean in the basis of sine waves, according to a standard Fourier decomposition. Couldn't we also decompose a signal into a different basis of orthogonal periodic functions, like square waves? According to a Fourier decomposition, a square wave has many frequencies. Indeed, when I hear a square wave, I can hear overtones. But in the basis of square waves, it has one frequency. Why is the Fourier decomposition more fundamental to human hearing?
 A: I refute the premise: we don't hear frequencies in the basis of sines. When we hear a sawtooth-like wave, we don't perceive it as a lot of superimposed sinusoidal partials, but rather as a single sound with a particular pitch and timbre. Only when you boost some individual partials strongly above the level of their neighbours, do they stand out as tones of their own right, though even then we ascribe the result rather to have a quality "like certain vocals", depending on where the boost lies. The wah-wah guitar effect is based around this phenomenon.
What is closer to true is to say Fourier basis provides a decomposition into information that we are sensitive to (amplitudes) and information we are not (or only hardly) sensitive to (phases). This can be witnessed by listening through music through all-pass filters, which scramble the phases of the partials but leave their amplitudes as they are. If you look at the waveform of an all-pass filtered signal, it appears completely different from the orginal, but if you listen to it it'll sound very similar.
It's pretty useful that the ears work this way, because phase information is completely unreliable in a natural environment: with delayed reflections from walls, wind dispersion etc., the same person speaking in two different rooms would be perceived as very different if we took phase into account. Different rooms do also change the amplitudes – sometimes a particular sine mode will phase-cancel between reflections – but most of the amplitude-ratios are generally similar between different acoustic spaces.
Now, the very name all-pass of course refers to the Fourier decomposition, to the fact that all sinusoidals come out with the same amplitude. But these filters can actually be defined without any reference to that at all:
All-pass filters are the operators under whose action any signal's autocorrelation stays invariant.
The autocorrelation of a signal $u$ is simply the $L^2$ correlation (i.e. inner product) with time-shifted versions of itself:
$$
  R_u(\tau) = \int_\mathbb{R}\!\mathrm{d}t\: u(t)\cdot u(t+\tau).
$$
It's easy to see why this is a good thing to look at if we want to be not sensitive to room acoustics: if you create a time-shifted version of the signal $\tilde u(t) = u(t-0.17\,\mathrm{s})$ (corresponding to sound reflection from a wall 30 metres away) then the autocorrelation comes out just the same:
$$\begin{align}
  R_{\tilde{u}}(\tau)
   =& \int\limits_{-\infty}^{\infty}\!\mathrm{d}t\: u(t-0.17\,\mathrm{s})\cdot u(t+\tau-0.17\,\mathrm{s})
   \\=& \int\limits_{-\infty+0.17\,\mathrm{s}}^{\infty+0.17\,\mathrm{s}}\!\mathrm{d}t\: u(t)\cdot u(t+\tau)
   = R_u(\tau).
\end{align}$$
What Fourier decomposition does have to do with this is that it can be used to calculate the autocorrelation efficiently. Observe that autocorrelation is convolution of the signal with a time-mirrored version of itself, and the Fourier transform the unique linear decomposition such that
$$
\mathcal{F}(\phi\star\psi)(\omega) = \mathcal{F}(\phi)(\omega) \cdot \mathcal{F}(\psi)(\omega).
$$
That's perhaps the most important reason why Fourier analysis is such a useful tool, though the point that march made about harmonic oscillators is certainly also relevant.
A: We can and do in some cases.  Take a look at the Zernike Polynomials for decomposing 2-dimensional frequency distributions, for example.
The short answer is that sine waves are nice and clean, behave well when applying Fourier or other transforms,  so why go make things difficult?
A: The "reason" that the Fourier decomposition is the "correct" one has to do with the fact that both signal detectors (microphones, ears, etc.) operate as driven harmonic oscillators. Or rather, it is the fact that the harmonic approximation tends to be a very good approximation for describing the kinematics of a objects that we use as detection devices.
If an object is a simple harmonic oscillator, then it has a natural frequency, and this frequency corresponds to sinusoidal motion of the oscillator. As a consequence, the simplest (and reasonable!) model for the coupling of a sound wave to the oscillator is one in which resonance occurs.  In this case, if the natural frequency of the oscillator appears in the Fourier expansion of the sound wave, then the oscillator will vibrate at this resonant frequency with an amplitude that depends directly on the amplitude of that Fourier component in the sound.  In this way, the Fourier components of the wave are "real" and therefore the Fourier expansion is a physically natural one to use.

Now, there is are a couple of huge caveats that go along with this.  The first is less important, so I'll deal with it first.  Direct detection objects like the diaphragms in microphones and the eardrum are very low-Q oscillators, which means that they don't really have a well-defined natural frequency.  They are in fact designed in such a way that they will respond relatively strongly across a broad frequency range, making them useful as detectors.  However, later on, there are often objects that have relatively well-defined natural frequencies, such as places on the basilar membrane in the inner ear.
The second problem is worse: the perception of sound is hugely complicated, not only because the signal processing going on in our brains involves non-linearities (such as feedback and feedforward between the processing centers of the brain and the sensory apparatus) but also because the mechanics of the detection itself in the inner ear is not so straight-forward as there being different objects that have well-defined natural frequencies.
As a first approximation, we can think of different places on the basilar membrane in the cochlea (where the nerves are that fire and send signals to the brain) having different well-defined natural frequencies of (sinusoidal) oscillation that individually respond to the different Fourier components of a wave traveling through the inner ear.  However, this simple picture is complicated by the fact that the modes are coupled to each other linearly due to the fact that the membrane is a membrane (there aren't actually isolated oscillators).  As a consequence, there is significant overlap between the different places on the membrane in terms of frequency response, especially at lower frequencies.
For this reason, the process of hearing involves at least two different mechanisms: the place theory that we've just described and the temporal theory, which doesn't involve a resonance phenomenon.  But that is a story for another question.
A: The vast majority of sound-conducting media are very well approximated by linear time-invariant systems: the response to added stimuli is the addition of the individual responses, and moving the stimulus in time will move the result in time.
LTI systems are completely characterised by their "impulse response".
The eigenfunctions of linear time-invariant systems are complex exponentials (essentially, sinoids with amplitude and phase shift): those functions as input to an LTI system are reproduced exactly, though with possibly differing phase and amplitude.
Other wave forms aren't.  The stacked impedance transformation and separation layers in the ear are also by and large LTI systems, and sound sources are usually well understood in terms of pulsed or synchronised excitations acting on resonating LTI systems.
If you start with a sine wave as signal, the arriving result will be a sine wave (snares are not LTI because the triggered contact noise is not linear in nature).  No other wave form is essentially unfazed by an LTI medium.  You cannot "muffle" a pure sine wave's sound, only dampen it.  It will not change character or clarity, only loudness.
In contrast, if you pass a triangular wave or a rectangular wave through a pillow, it will change character.
Because sine waves are such exceptionally important constituting invariants in any soundscape, the hearing is well-tuned to dealing with them, partly by the construction of the whole inner ear, but also by the resulting conversion to nerve pulse trains (which are not linear phenomena but able to represent them comparatively well) from spatially separated detectors and the interpretive layers in the cortical processing.
A: To put your question in a wider perspective: consider white noise
As we know: the expression 'white noise' is used for continuous sound that consists of random frequencies; random shifting of frequency, random shifting of the  volume of each consisting frequency (resulting in a roughly  continuous volume for the overall white noise).
In the case of white noise any decomposition is useless.
That is: in the case of white noise decomposition into a basis of sine waves is as useless as any other decomposition into some basis. So: in the case of white noise fourier decomposition into a a basis of sines is not a preferred decomposition.

Music instruments and resonance
Music instruments that are build to a tried and tested design have the property that the sound that is produced carries well.
Early versions of that instrment may have had a sound that carried less well, and those versions were abandoned. The tried and tested designs developed over centuries, always with an ear for producing sound that carries well.
A flute is designed to have good resonance at every pitch of its range. Of course: by opening and closing specific holes/valves the instrument is reconfigured to resonate at the pitch that the flute player is producing.
Resonance is always best when the vibrating medium has optimal opportunity to oscillate in harmonic oscillation.
So: even when instrument design developed through mostly trial and error, the design very much tends to favor producing sound that consists of harmonic oscillations that are multiples of the base frequency that the instrument is producing.

The psychological dimension is that we humans are far more interested in sounds that consist of multiple harmonics.
White noise is totally uninteresting; we cannot hear anything in it.
It is sounds with discernible internal structure that interest us the most.
For the sounds that are the most interesting to our human auditory perception: Fourier decomposition into a basis of sines is in effect the only option.
A: The human sensory system is a large ensemble of detectors which are made of nerve cells.  A complex sensation occurs when a stimulus triggers a large number of nerve cells, while a simple sensation triggers a small number of nerve cells.
For a “touch” example, lay a pencil flat in the palm of your hand. You feel the pencil near the wrist, and also near the fingers, and in between. Now take that pencil and poke your palm with the point. You are triggering a smaller number of sensory cells, so the sensation feels more “pure.”
The palm of your hand has a high density of touch sensors, but your back has a much lower density.  There is a well-known party trick where you touch someone of the back with the points of an unknown number of pencils: maybe just one, or maybe four or six held together in your fist.  The different pencil points don’t activate different sensory nerves in your back, so the person being poked can’t distinguish between one point versus several points.
In your eye, the lens of your eye takes light rays from a particular direction and focuses them onto a particular location on your retina, where there is a particular nerve cell waiting to sense them.
In your ear, the nerve cells line the cochlea, which is a fluid-filled tapered tube.  A “pure” tone excites some of these nerve cells without exciting others.  I am a little murky on exactly how this happens. I have read in the past that the tapered shape of the cochlea sets up a wavelength-dependent standing wave with an antinode at the nerve cells corresponding to a particular pitch.  As a physicist, I get excited about the idea of resonant amplification playing a role in biology. However, this antinode hypothesis seems inconsistent with the fact that high frequencies are detected at the wide mouth of the cochlea, and lower frequencies deeper in.  Another possibility is that sound waves in the cochlear fluid undergo frequency-dependent extinction, and that the sensation of a “pure” pitch corresponds to a well-defined boundary between upstream cochlear cells which are excited versus downstream cells which are idle.
This extinction-based model of pitch sensitivity explains why loud low-frequency sounds accelerate age-related high-frequency hearing loss.  It also moves the question of sinusoidal “pure” tones from biology and neurology to the domain of frequency analysis of damped oscillations in fluids.
