I tend to use Macaulay2 for research related stuff
Some points about it.
- It is open source
- It is specific to algebraic geometry/commutative algebra (but since this is a large community on mathoverflow...)
- Is actively being developed (it seems to have several conferences devoted to it each year in recent years).
- It is a unix program so you need cygwin to run it on windows.
- The preferred environment to run it seems to be in an emacs window (although various people are working on other interfaces I understand).
I've also heard good things about Sage but haven't tried it.
