For antivirus programs, you should only run one at a time. For optimum security, some people like to have two or even three installed, and that's fine. The problem occurrs when two are running at once...they can either freeze as two try to cleanse the same file at the same time, or even identify eachother as threatening and try to remove eachother. Thus, you have to minipulate them carefully. Set one as the always-on scanner, and make sure the others don't launch at startup. Then, if you want a ''second opinion,'' shut down the one completely, then launch the second antivirus program.
A similar situation exists with antispyware software. With these utilities, you can usually run several real-time scanners at once without a problem. (I have Microsoft Antispyware, SpyBot's TeaTimer, and Spyware Blaster all set for real-time scanning.) However, it's recommended to run each full system scan seperately. (For instance, I have Microsoft Antispyware to automatically run a full scan daily, and once it's finished, I run SpyBot, then AdAware, then...) If they all run scans (not just open, but actually scanning) at the same time, it can cause your computer to slow down to a crawl, overtax your hard drive, and cause a conflict if two utilities try to scan/delete the same file at the same time.
In short, it's recommended to have multiple adware/spyware scanners installed, but try not to have them all set to scan at the same time. And, when speaking of firewalls and antivirus software, unless you're extremely concerned about security (and know how to manipulate multiple security programs properly), stick to one of each. (I use ZoneAlarm Professional firewall and Avast 4 Professional antivirus, but free versions are also available here and here.)
Hope this helps,
P.S.: Like I mentioned in the other post, AdAware scans for adware, not spyware. While the two types of invaders tend to overlap at times, you need both adware and spyware scanners installed and running.