What is the speed of sound in space?

By popular demand (considering two to be popular — thanks @Rod Vance and @Love Learning), I'll expand a bit on my comment to @Kieran Hunt's answer:

Thermal equilibrium

As I said in the comment, the notion of sound in space plays a very significant role in cosmology: When the Universe was very young, dark matter, normal ("baryonic") matter, and light (photons) was in thermal equilibrium, i.e. they shared the same (average) energy per particle, or temperature. This temperature was so high, that neutral atoms couldn't form; any electron caught by a proton would soon be knocked off by a photon (or another particle). The photons themselves couldn't travel very far, before hitting a free electron.

Speed of sound in the primordial soup

Everything was very smooth, no galaxies or anything like that had formed. Stuff was still slightly clumpy, though, and the clumps grew in size due to gravity. But as a clump grows, pressure from baryons and photons increase, counteracting the collapse, and pushing baryons and photons outwards, while the dark matter tends to stay at the center of the overdensity, since it doesn't care about pressure. This creates oscillations, or sound waves with tremendously long wavelengths.

For a photon gas, the speed of sound is $$ \begin{array}{rcl} c_\mathrm{s} & = & \sqrt{p/\rho} \\ & = & \sqrt{c^2/3} \\ & \simeq & 0.58c, \end{array} $$ where $c$ is the speed of light, and $p$ and $\rho$ are the pressure and density of the gas. In other words, the speed of sound at that time was more than half the speed of light (for high temperatures there is a small correction to this of order $10^{-5}$; Partovi 1994).

In a non-relativistic medium, the speed of sound is $c_\mathrm{s} = \sqrt{\partial p / \partial \rho}$, which for an ideal gas reduces to the formula given by @Kieran Hunt. Although in outer space both $p$ and $\rho$ are extremely small, there $are$ particles and hence it odes make sense to talk about speed of sound in space. Depending on the environment, it typically evaluates to many kilometers per second (i.e. much higher than on Earth, but much, much smaller than in the early Universe).

Recombination and decoupling

As the Universe expanded, it gradually cooled down. At an age of roughly 200,000 years it had reached a temperature of ~4000 K, and protons and electrons started being able to combine to form neutral atoms without immediately being ionized again. This is called the "Epoch of Recombination", though they hadn't previously been combined.

At ~380,000 years, when the temperature was ~3000 K, most of the Universe was neutral. With the free electrons gone, photons could now stream freely, diffusing away and relieving the overdensity of its pressure. The photons are said to decouple from the baryons.

Cosmic microwave background

The radiation that decoupled has ever since redshifted due to the expansion of the Universe, and since the Universe has now expanded ~1100 times, we see the light (called the cosmic microwave background, or CMB) not with a temperature of 3000 K (which was the temperature of the Universe at the time of decoupling), but a temperature of (3000 K)/1100 = 2.73 K, which is the temperature that @Kieran Hunt refers to in his answer.

Baryon acoustic oscillations

These overdensities, or baryon acoustic oscillations (BAOs), exist on much larger scales than galaxies, but galaxies tend to clump on these scales, which has ever since expanded and now has a characteristic scale of ~100 $h^{-1}$Mpc, or 465 million lightyears. Measuring how the inter-clump distance change with time provides a way of understanding the expansion history, and acceleration, of the Universe, independent of other methods such as supernovae and the CMB. And beautifully, the methods all agree.

From the ideal gas law, we know: $$ v_\textrm{sound} = \sqrt{\frac{\gamma k_\textrm{B} T}{m}} $$ Assuming that interstellar space is heated uniformly by the CMB, it will have a temperature of $2.73\ \mathrm{K}$. We know that most of this medium comprises protons and neutral hydrogen atoms at a density of about 1 atom/cm⁻³. This means that $\gamma = 5/3$, and $m = 1.66\times 10^{-27}\ \mathrm{kg}$, giving a value for $v_\textrm{sound}$ of $192\ \mathrm{m\ s^{-1}}$.

However, this is not propagated efficiently in a vacuum. In the extremely high vacuum of outer space, the mean free path is millions of kilometres, so any particle lucky enough* to be in contact with the sound-producing object would have to travel light-seconds before being able to impart that information in a secondary collision.

*Which for the density given, would only be about 50 hydrogen atoms if you clapped your hands – very low sound power!

-Edit- As has quite rightly been pointed out in the comments, the interstellar medium is not that cold. At the moment, our solar system is moving through a cloud of gas at approximately 6000 K. At this temperature, the speed of sound would be approximately $9000\ \mathrm{m\ s^{-1}}$.

See Kyle's answer for a table of values for $v_\textrm{sound}$ that can be found in different environments in space, or pela's for information on how early universe sound waves became responsible for modern-day large scale structure.

Just want to bring up that most answers seem to be taking "space" to be a nice uniform medium. However, even within our own galaxy, conditions vary wildly. Here are the most common environments in the Milky Way:

• Molecular Clouds, $\rho\sim 10^4\,{\rm atom}/{\rm cm}^3$, $T\sim 10\,{\rm K}$
• Cold Neutral Medium, $\rho\sim 20\,{\rm atom}/{\rm cm}^3$, $T\sim 100\,{\rm K}$
• Warm Neutral Medium, $\rho\sim 0.5\,{\rm atom}/{\rm cm}^3$, $T\sim 10^4\,{\rm K}$
• Warm Ionized Medium, $\rho\sim 0.5\,{\rm atom}/{\rm cm}^3$, $T\sim 8000\,{\rm K}$
• HII Region, $\rho\sim 1000\,{\rm atom}/{\rm cm}^3$, $T\sim 8000\,{\rm K}$
• Hot Ionized Medium, $\rho\sim 10^{-3}\,{\rm atom}/{\rm cm}^3$, $T\sim \;{>}10^6\,{\rm K}$

The sound speed is proportional to $\sqrt{T}$. Given that the temperature varies over about 7 orders of magnitude (maximum at about $10^7\,{\rm K}$, minimum at about $3\,{\rm K}$), the sound speed varies by at least a factor of $1000$. The sound speed in a warm region is on the order of $10\,{\rm km}/{\rm s}$.

Trivia: the sound speed plays a crucial role in many astrophysical processes. This speed defines the time it takes for a pressure wave to propagate a given distance. One place this is a key time scale is in gravitational collapse. If the sound crossing time for a gas cloud exceeds the gravitational free fall time (time for a gravity-driven disturbance to propagate), pressure is unable to resist gravitational collapse and the cloud is headed toward the creation of a more compact object (denser cloud, or if conditions are right, a star).

More trivia: space is a very poor carrier (non carrier) of high frequency sounds because the highest frequency pressure wave that can be transmitted has a wavelength of about the mean free path (MFP) of gas particles. The MFP in space is large, so the frequency limit is low.