On a Sunday when a Later Day Saint's Church meets, what roles are women allowed or not allowed to play in the service?
Well, so the big hidden question here is "what comprises an LDS sunday service?". A related question covers some of this, so I'll just summarize. In your average "Sacrament meeting", you'll typically have:
Of these, the only thing women may not do is preside at the meeting, bless the Sacrament, or pass the Sacrament. Those are duties of people who hold the Priesthood, and only men hold the Priesthood.
It would be very unusual to see a woman conduct the meeting, since that is usually done by the presiding person or one of his counselors. However, there are some meetings of the Church where women conduct the meetings (for example, the General Relief Society Meeting). There are also meetings where women preside (again, most meetings of the Relief Society), but that does not happen in a Sacrament meeting.
The LDS website gives a very succinct answer to this:
The page linked to above contains many links to more information, but in short, women hold an important place and serve as teachers and role models, and are in authority in certain positions.
There are positions they cannot hold: most significantly, the priesthood. From the BYU's wiki:
The duties of the Priesthood are covered here, which gives some insight, but not specifics of exactly what women are, and are not allowed to do.