First of all, viruses are quite different from spyware, so even at a conceptual level, there shouldn't be a conflict.
But, more specifically, I'm not aware of problems arising, even if you run two or more products within each category.
I even think it's advisable to run two or more spyware programs. I use Ad-Aware SE and AVG Anti-Spyware, and have found consistently that one will find problem files that the other one missed (and it works both ways).
For a web site that provide a basic overview of internet security issues, try...
http://Confidential-Transactions.com