Don't know if you can get the exact number, but there are a couple of ways that can give you an idea:
On .bomb, if it is more than 3 hours before departure, you can log in, and go through the steps as if you are booking a ticket. If you're in expert mode, when the flights come up, under flight details, a list of booking classes and the number of seats remaining to be sold will show up. You can also go to seatcounter (and flightstats has a way to see this too, now), where you can plug in your route and time of day (and preferred airline), and see this in the same general format. Use Y for the economy cabin, as these are the highest price tickets sold, so these will be the best representation of what they're willing to sell. A 9 means that there are at least 9 seats available for sale still (i.e. 9 or more). Anything less means there are that number of seats left. Note, due to oversell's, no show's etc. these numbers aren't 100% accurate, but they give you an idea.
I've also always gotten answers from phone agents on the day of/before travel. Sometimes I'll call and say I'm looking to standby for an earlier flight between XXX and YYY, or I'm looking to standby on flight XXXX, and was wondering if there's a good chance to be accomodated. Often, I am actually trying to standby. They won't give you exact numbers, but if they tell me yes, I'll often ask if there are a few, or many seats left, and if they say not really, I'll ask if there are still a couple of seats available, or if its actually sold out/oversold, and they usually are able to give you that information. Again, not exact, but it gives you a good idea.