Yeah my wife and I went through this with JSA. I am a British Citizen and entitled to it, and need it at the moment since I was made redundant. You have to notify the council that your partner is living with you and they will give you a form to fill out to make a couple's claim you cannot claim single because it is benefits fraud.
What they did in our case was when I filled out the form we informed them my wife's visa restricted her and that she is not allowed to receive public funds or work. They sent her to a N.I interview and they made copies of her passport, our marriage certificate and reasons for her needing an N.I and refused her one based on her not requiring one since she cannot work.
The job centre plus was notified of this, and notified of why she was refused the N.I and it is now on their system so they scrapped my couple's claim, added her info to my singles claim and kept me on singles rate. With child tax credits, council tax and housing benefit, all they want to know is that there is proof of your partner's identity which you can prove by sending copies of the N.I application acceptance, and/or the refusal letter for an N.I and in the case of housing benefit, the council website and HMRC states that if one partner has an N.I and is eligible for benefit but if the other partner is not eligible or does not have an N.I they are ignored or treated as having an N.I anyways.
If your partner is subject to immigration control, and you make a joint claim for any benefit they will set up a N.I interview, at the interview they will check the visa status of the partner and if they are ineligible for work or public funds you'll get a letter in the post stating they are not allocated a N.I and you send a copy of that letter to the council and they will accept your claim in most cases.
There is a form called DCI1B that the local authorities send to the N.I office that is for those who cannot get an N.I but since wife and I are legally married, I am not sure how these rules work with unmarried couples and I could imagine it being a lot more difficult in those situations. Do not claim as a single when living with a partner under any circumstance because they will eventually find out and you will end up losing your benefit and can get in trouble for fraud. It is better to be safe than sorry and go through all the hoops and loops and let them decide to keep your claim single rather then claiming single yourself. Depending on the circumstances they may award you couple's rate or they may not, it is up to the local authorities to decide what would be in best interest.
Here are some helpful links:
http://www.housing-rights.info/03_1_2_H ... it_law.php
http://www.housing-rights.info/housing- ... otland.php
http://www.housing-rights.info/index.php
http://www.hmrc.gov.uk/taxcredits/start ... -claim.htm
http://www.hmrc.gov.uk/taxcredits/start ... -in-uk.htm