For mosques, if you are a guy, wear long pants (full-length). Wash your feet before you step into the mosque proper. They have the washing area for the muslims to do abolution (cleaning up) before they do the prayers. Avoid the prayer times. As for prayer times, you can check this link
http://www.muis.gov.sg/cms/uploadedFiles/MuisGovSG/Downloads/Prayer%20Timetable%202010.pdf. And of course, avoid Friday afternoon, as all Muslims will be gathering for their Friday prayers. If you'd like to shoot such a congregation, you can ask the Imam of the mosque.
As for women, would have to cover up with a scarf, long sleeves blouse and long pants (full length). Again, wash your feet before entering the mosque proper. They have different washing areas for men and women, so please take note.
Before entering, check and ask the Imam (mosque leader) about more detailed etiquettes. They are more than happy to invite you in and explain to you.
One last thing, remove your shoes prior to entering the mosque. Removing the socks is also good as you would also know you'd have to wash your feet.
HTH.