Whilst he may appear to have another address, an investigator would look at where his credit is, if he's a payer on utility bills, where his car is registered and where it's parked during the week. If he had even one credit card or insurance policy or mobile phone linked to her address, it's grounds to be questioned. If, when he filled in forms, he put her address, it'd count against him. Besides claiming as a single mother, she's undoubtedly committing discount fraud. Not that I think either of them are clever enough to register to vote, but if he was on the electoral register at her address whilst she claimed a single occupancy discount, she'd be questioned. Most fraud occurs from people reporting suspicions. If you have suspicions, report her. They'd look at social media as well.