Look at the [Survey of Doctoral Recipients](http://www.nsf.gov/statistics/nsf09317/content.cfm?pub_id=3920&id=2).  This is a large scale survey of US doctoral recipients.  I'm not sure how much they've avoided sample bias.  They have 29,170 employed mathematicians in their sample, of which 

* 17,290 are employed at "universities and 4-year colleges."
* 860 are employed at "Other educational institutions"
* 7,310 are employed at private companies
* 1,050 at non-profits
* 1,290 by the federal government
* 350 by local governments
* 1,020 are self-employed

Of those employed at "universities and 4-year colleges"

* 5,340 graduated less than 10 years before the survey, of whom 880 are tenured,  2,750 are tenure track, 390 not on tenure track, and 1,310 are in positions where tenure is not applicable (it looks like 1,080 of these are postdocs)
* 11,950 graduated more than 10 years before the survey, of whom 9,920 are tenured,  520 are tenure track, 730 not on tenure track, and 780 are in positions where tenure is not applicable (again, not sure what that means)

The total number in the sample who got their Ph.D.s more than 10 before the survey was 19,790, so about half of them got tenured positions at universities and four year colleges (of course, that number will go up a lot when you remove people who left academia right after graduating).  

[Edit by Stephen Griffeth]:

The survey by the NSF is of "doctoral scientists and engineers": their definition is 

"Doctoral scientists and engineers are defined in this report as individuals less than 76 years of age who have received a doctorate in a science, engineering, or health field from a U.S. academic institution and who resided in the United States or one of its territories on 1 April 2006."

So Yiftach's objection that the market is really international has force here.  And a Ben notes, maybe there's some sample bias (perhaps people who hadn't done so well did not return the survey?) On the other hand, I still think I got some useful information out of the link Ben provides above.  

For instance, of the 33,830 people included in the survey with a math/stats PhD, about 4000 were retired, 330 unemployed, another 330 neither employed nor seeking work, and of the remaining 29000 about 26000 were employed full-time.  The "involutary out-of-field rate" was 4% (lower than the unemployment rate for just about every other type of degree!).  I wonder whether we can safely assume that this gives an upper bound on the unemployment rate for US math/stats PhDs (most will not leave the country without work---or are there enough internationals with visa issues to balance this out?)

Another useful piece of information: the number of math/stats PhDs in the US is not keeping up with population growth.  Of the roughly 29000 *employed* PhDs in the sample, about 8200 were between the ages of 55 and 64, about 6400 were between 45 and 54, about 8200 were between 35 and 44, and about 3300 were less than 34.

Of the 17,290 math/stats PhDs employed by "universities and 4-year colleges", about 5000 are primarily researchers, while about 10000 are primarily teachers, with another 2000 or so in other capacities (most popularly, "management/sales/administration").  It's hard for me to tell how many of the 5000 in research have tenure---though apparently the number math/stats PhDs with "postdocs" is about 1160 (I'm not sure why this differs with Ben's number above---mine is from table 36), with 660 "mathematical scientists" and 500 "postsecondary teachers" (I don't know what this means... perhaps the mathematical scientists are the ones who are primarily researchers?)

Unfortunately, I still don't know the answer to the question that motivated me (*given* that you get a postdoc after grad school, what's the probability that you wind up as a tenured research mathematician?)  But my feeling is that this probability is much higher than implied by the grim discussion in Philip Greenspun's article (linked to in Igor's answer to Ben's question, that I linked to above).