Hi Zola,
There's a running track at School of Physical Education (SPE) next to the National Institute of Education, near the Bukit Timah entrance of the Botanic Gardens. The track is an ok one.
As for womens rugby: there are 2 full contact and about 8-10 touch rugby teams. If you're interested in touch rugby, then several teams train across the island, my team Bedok Kings trains at SPE for pre- and during season (june-november). We also do "keep fit" touch rugby out-of-season on saturdays down the the People Association off Nichol Highway, near the National Stadium. (This is mixed).
Don't know much about the full-contact, I think Blacks and Bucks both have teams.
Contact me if you want more details on the rugby!
GO THE KINGS!!!