Here's a little more detail about timing of math postdocs offers, since that seems to be a particular focus of this question. What I'll describe applies only to research-oriented postdoctoral positions offered by mathematics departments in the U.S. Tenure-track jobs and anything teaching-focused are on a somewhat different schedule, and positions outside the U.S. may be totally different.
Everything is synchronized by the AMS common deadline, because most or all of the top candidates (depending on how strict your definition of "top" is) will have accepted offers by then. In particular, just about every position will have been offered to someone by then, but many of those offers will be turned down. Most departments will not set an earlier deadline than the AMS deadline, but they nevertheless encourage earlier decisions if possible. When a position is turned down, a competent department will try to make another offer as quickly as they can. Who gets the next offer depends not just on relative rankings of candidates, but also on factors such as research specialty and game-theoretic issues such as perceived likelihood of accepting the offer. Administrative dysfunction may limit how quickly offers can be made, but it's not uncommon for a department to go through several rounds of offers in January.
In practice, it seems to work out more or less as follows:
A few people get offers in December, but this is rather unusual. You shouldn't worry at all if you have no offers as of January 1.
The people perceived as top candidates generally start to get offers in early January. If you make it a couple of weeks into January with no offers, then you probably aren't going to end up getting tons of offers, but you shouldn't feel discouraged. All you need is one offer you're happy with.
As January progresses, more and more offers are made. Quite a few candidates, but certainly not all, will get an offer in the second half of January.
If you do get an offer, you should immediately do two things. First, you should withdraw any applications to (or turn down offers from) places you are no longer interested in, so they don't waste time and so you don't tie up offers someone else is waiting for. It's reasonable to hold onto several offers at once if you genuinely can't decide between them, but you do this only when truly necessary. Second, you should write to any departments you would prefer to your current offers to let them know about your upcoming deadline. The competition may increase their interest in you, and in any case they need to know they'll miss their chance at you if they don't move fast enough.
If you haven't yet received an offer, the second half of January is a good time to make inquiries. You can do this yourself, but it's sometimes better if your advisor looks into it. (All you can do is express interest, but your advisor can put in a good word for you and try to find out how things are going through back channels.)
As the AMS deadline approaches, there's a flurry of activity. Many offers are turned down or accepted around then, and departments are eager to get their favorite candidates before they accept another job, so there's a lot of turn-over for offers.
If you make it past the first week in February without an offer, it's a bad sign. It's certainly not a disaster, and plenty of strong mathematicians have gotten offers later than that. However, this is the point at which you need to start taking action, with your advisor's help. You need to figure out which positions are still open, and make sure they are aware that you are still interested. It's really useful if your advisor can work his/her contacts to help figure out what might still be a possibility and to make sure you don't fall through the cracks.
The more times passes, the more you need to actively search for a job, rather than just waiting to see what happens. If you hit March, then you're in trouble. You can still find a job then, but you need to be open to possibilities you weren't originally hoping for. For example, there are often late job postings for unpredictable openings, such as a temporary replacement for someone on sabbatical or leave. In March, you should be applying for every new opening you would be willing to take (and you should have a backup plan for what happens if you don't get a job at all, to help you decide where to draw the line). This is also a good time to review options with your advisor. Could you delay graduation for a year, or spend the next year as a lecturer?
I don't mean for this to sound discouraging. It's certainly possible to get a job quite late in the season, and you might even get lucky and find a really good job. However, I think it's worth having an overview of what the plausible outcomes are at each stage.
TL;DR: If you are hoping for a research-oriented postdoc in math, don't worry until late January, worry somewhat in February, and worry a lot in March (but don't panic).
