Does a "home based" DSL modem use one static IP ?

depends if you buy a package that has one or not.

Ouch! Not much information here.

Your DSL modem gets an external (world-wide) IP number from your ISP.
How this is allocated depends on your ISP: Some will allocate you a "static address" that will always stay the same, forever, as long as you pay the subscription. This generally costs more, and is currently quite unusual. When IPv6 comes in, I expect this'll be the norm, and cheap, but this hasn't happened yet.

Other ISPs will allocate you a "dynamic address" which is allocated when you boot up your DSL modem and establish a connection with your ISP. The (external, world,) address may change as your ISP sees fit, but you'll get an address and it will work for emails and browsing. It may not work if you wish to connect to your computer from elsewhere, because your allocated IP address may have changed.

If you want to know what your external IP address is you can:
1] Pay for a static address (contact your ISP about this)
2] Go to http://whatismyip.com and find out what your address currently is.
3] Go to http://www.no-ip.com/ and sign up.

Thanks acid kewpie and tredegar, no, I did not buy a static IP but if the DSL modem stays on and connected full time to the ISP even if the home computer is turned off, does the DSL modem retain the initially assigned IP or does the ISP send out a new IP periodically ?

It can happen either way, My ISP forces a new IP address every 24 hours..
They do it intentionally to make it annoying for you to host any type of server on your connection..

ISP's vary on this one..

Thanks farslayer, that supports my experience.