If you suspect a fraud is taking place I would strongly advise you not to submit a false claim for housing benefit as some sort of test.
Tenants are responsible for paying council tax however this responsibility can be contracted to the landlord instead, your contract would help in any court case but the council would chase you (as the tenant) in the first instance.
You run the risk of causing yourself a lot of grief, but if you have found yourself in the middle of a fraud it might be best to do something about it as soon as possible. A trip to the council office to discuss council tax etc would probably be the place to start. Also you should register to vote from that address, that would also flag the issue to the council.
If your landlord has not taken out a mortgage that allows the property to be rented then she could be trying to avoid the mortgage company finding out by paying your council tax and water rates? might not be about benefit fraud.