I'm about to enter a math PhD program at a pretty good school, but the job market for research university professors has been terrible as far as I can remember (since the late 2000s recession). I hear many stories of hundreds of excellent candidates applying for one or two openings.
Older professors tell me it wasn't always like this. What did the market look like when it was considered "good" and could it ever come back by the time I finish in 5-6 years? Or has there been a fundamental shift in academia preventing the good times as we knew it from ever returning?
Compare page 3 of the AMS annual survey from 1999 with page 4 of the 2008 edition and page 4 of the 2014 edition, in terms of employment for new hires we are actually much better currently than the early 90s.
Depending on the age of the individual you talked to: the good old days could be everything up until the late 80s, when rate of unemployment remain largely below 2%, and new graduates are often immediately hired on tenure-track positions. We are almost certainly not going to return to that level of employment: in the post-WWII era through the cold war there have been (for one reason or another, GI Bill for example) expansions in funding for academia. That growth was never sustainable. As a benchmark for the future you probably want to consider the data from 1990 on.
If you look in the 1999 data, of those obtaining employment in the United States, just under 30% (of the new PhD recipients) are hired in postdoc positions and a bit over 50% in permanent positions (30% in academia and the other 20% in industry/government). The 2008 data is similar. The postdoc percentage is higher in 2014 (38%) but not outrageously so.
You can spend some more time teasing out the data yourself; the annual survey results are available online for the past 15 years, and presumably the AMS has the old data dating back to the 50s available somewhere.
If you want to be more forward looking and consider hiring statistics overall instead of just for new PhDs, since the economic crisis the AMS has been running employment surveys every years. A quick glance at the 2014 survey suggests that things have improved slightly.
As a side remark: the large number of applications mathematics positions receive has in some part to do with the availability of MathJobs.org. The website makes applying to 50-100 positions in one hiring cycle relatively easier in mathematics compared to other academic fields without a similar centralized job database. So the inflation of number of applications received per opening compared to "old days" does not 100% correlate to the health of the job market.