Does every statistic have a sampling distribution, not just the sample mean? I am curious because most basic undergraduate statistics reference just start out Inferential Statistics by mentioning sampling distributions and the sampling distribution of the mean. My question is that does every statistic have one? Even the sample proportion, sample variance and standard deviation?
Edit: What would they be like?
 A: Yes, as every statistic is a function of you sample (which are random variables) they will have a distribution. It might not be as easy to deduce the distribution as with the sample mean.
A: Yes, every statistic has a sampling distribution (though some may be degenerate).

What would they be like?

The sampling distribution of a statistic - just as with the mean - will in general depend on the population distribution you start with (and the sample size, naturally).
As an example, in a random sample from a normal distribution, the sample variance is a multiple of a chi-squared random variable and so the sample s.d. is a multiple of a chi random variable.
Below is a histogram of the sample standard deviations from 10000 samples of size 10 from a normal distribution, and the true sampling distribution (scaled-chi, red curve):

(click for larger version)
If you don't start with a normal population, the distribution of the sample s.d. is something else. E.g. here's the sample sd for 10000 samples of size 10 from a uniform distribution:

As we see, this one is mildly left skew rather than mildly right skew (I didn't calculate its theoretical distribution).
Note also that a sample proportion is a form of mean (label the in-category observations with a 1 and the out-of-category observations with a 0 and the sample mean is the sample proportion you started with). If the probability of being in group is constant and the observations are independent, it will have a discrete sampling distribution; a scaled binomial.
Many statistics are asymptotically normal under fairly mild conditions, but many are not (e.g. consider sample maxima for one).
Sampling distributions of various statistics come up in a number of situations. As an example, sampling distributions are important in hypothesis testing.
