The other day, I bumped my bookshelf and a coin fell down. This gave me an idea. Is it possible to compute the mass of a coin, based on the sound emitted when it falls?
I think that there should be a way to do it. But how?

So, I decided to try it out. I used Audacity to record ~5 seconds of sound that resulted when I dropped a penny, nickel, dime, and quarter onto my table, each 10 times.  I then computed the power spectral density of the sound and obtained the following results:

I also recorded 5 seconds of me not dropping a coin 10 times to get a background measurement.  In the plot, I've plotted all 50 traces on top of one another with each line being semi-transparent.
There are several features worth noticing.  First, there are some very distinct peaks, namely the 16 kHz and 9 kHz quarter spikes, as well as the 14 kHz nickel spike.  But, it doesn't appear as though the frequencies follow any simple relationship like the $ \propto m^{-1/3}$ scaling the order of magnitude result Floris suggests.
But, I had another idea.  For the most part, we could make the gross assumption that the total energy radiated away as sound would be a fixed fraction of the total energy of the collision. The precise details of the fraction radiated as sound would surely depend on a lot of variables outside our control in detail, but for the most part, for a set of standard coins (which are all various, similar, metals), and a given table, I would expect this fraction to be fairly constant.
Since the energy of a coin, if it's falling from a fixed height, is proportional to its mass, I would expect the sound energy to be proportional to its mass as well.  So, this is what I did.  I integrated the power spectral densities and fit them into a linear relationship with respect to the mass.  I obtained:

I did a Bayesian fit to get an estimate of the errors.  On the left, I'm plotting the joint posterior probability distribution for the $\alpha$ intercept parameter and the $\beta$ slope parameter, and on the right, I'm plotting the best fit line, as well as $2\sigma$ contours around it to either side.  For my priors, I took Jeffrey's priors.
The model seems to do fairly well, so assuming you knew the height that coins were dropping and had already calibrated to the particular table and noise conditions in the room under consideration, it would appear as though, from a recording of the sound the coin made as it fell, you could expect to estimate the mass of the coin to within about a 2-gram window.
For specificity, I used the following coins:

Penny: 1970
Nickel: 1999
Dime: 1991
Quarter: 1995

Edit: Scaling Collapse
Following Floris, we can check to see how accurate the model $ f \sim E^{1/2} m^{-1/3} \eta^{-1} $ is.  We will use the data provided, and plot our observed power density versus a scaled frequency $f m^{1/3} \eta E^{-1/2}$.  We obtain:

which looks pretty good.  In order to see a little better how well they overlap, I will reproduce the plot but introduce an offset between each of the coins:

It is pretty impressive how well the spectra line up.  As for the secondary peaks for the quarter and nickel, see Floris' afterthought.
Landing Material
Someone in the comments asked what happens if we change the thing the coins fall onto.  So, I did some drops where instead of falling onto the table directly, I had the coins fall onto a piece of paper on the table.  If you ask me, these two cases sounded very different, but their spectra are very similar.  This was for the quarter.  You'll notice that the paper traces are noticeably below the table ones.

Coin Materials
The actual composition of the coin seems to have a fairly large effect.  Next, I tried three different pennies, each dropped 5 times.  A 1970s brass penny, A 2013 zinc penny and a 1956 bronze penny.

Large Coins
Hoping to better resolve the second harmonic, I tried some other larger coins:

Notice that the presidential dollar has a nicely resolved second harmonic.  Notice also that the Susan B dollars not only look and feel like quarters, they sound like them too.
Repeatability
Lastly, I worried about just how repeatable this all was.  Could you actually hope to measure some of these spectra and then given any sound of a coin falling determine which coins were present, or perhaps as in spectroscopy tell the ratios of coins present in the fall.  The last thing I tried was to drop 10 pennies at once, and 10 nickels at once to see how well resolved the spectra were.

While it is fair to say that we can still nicely resolve the penny peak, it seems nickels in the real world have a lot of variations.  For more on nickels, see Floris' second answer.
