I think a difficulty is that you'll have a problem making a case you are effectively married, if you haven't been living together.
Or to put it another way, your plans, marriage plans, mean nothing to the Singapore authorities. What they see is a boyfriend wanting to get resident's status on the back of his girlfriend.
Unless you prove otherwise.
Edit to add: This was written before I saw your follow-up of 7:18
What embassy are you thinking will attest to your common-law marriage? I wonder if India even recognises the idea. It surely doesn't matter if where you live recognises it, as you are not American. And Britain, in my experience they are loath to stray outside of their standard embassy services. Attesting to something in America that they can't possibly have any knowledge of not being one of them.
If you have a persuasive case it can be done. But unfortunately people who do it never seem to come back to the forum after taking advice, and let us know how they did it. Perhaps you will be different, though odds on probably not.
Maybe try contacting this poster from yesterday and seeing if she can shed any light for you...
http://forum.singaporeexpats.com/sutra5 ... ht=#594114