Ok. I think we definitely have guests living with us. But we have virtually no evidence of them coming into the house rooms to eat or crap or whatever. And therefore I can't work out what on earth to do about catching or killing them. 
So several weeks ago, we heard a scrabbling sound up at the top of the living room wall, sort of 'behind' the coving. The cat (dopey bugger that he is) heard it to and knocked stuff off the mantelpiece repeatedly, trying to get close to it. But we never saw any evidence of rodents coming into rooms and eating. A couple of weeks after we started hearing this noise, and it only happening intermittently, I heard a gnawing right beside my bed, where I was keeping bags of Christmas presents for the children. The cat heard it too and had a good rubberneck. 
The next morning, I looked at the bags and realised that a mouse had come into the bedroom and gnawed a chocolate coin (stocking stash)! This remains the only time we've ever had evidence of a mouse/rat in a room, eating our food.
Then, a couple of days after the coin incident, nothing, for weeks and weeks. I assumed the dopey cat had dropped a shrew or something, which wasn't well adapted to human food and allergic to chocolate or something. And now the scrabbling noises have started up again. What is worse, I heard actual squeaking up in the walls, which surely means it's more than one. Yet still no evidence of gnawed food, etc. No poo, no stink. We hear something quite weighty skittering across the ceiling (i.e. under the upstairs floorboards) in the evenings now. But no poo, no food eaten!
So what can I do? Clearly we have a problem or are about to have one. We're a 1920 semi and the house adjoining was empty for over a year until this summer - prior to that, it was lived in by a very old man with dementia and extremely low personal standards of hygiene 
who couldn't cope on his own. If he had a mouse infestation I wouldn't be surprised - he probably wouldn't have even noticed. Could it be that the mice/rats are still next door, being PITAs and eating our neighbours' food and just coming through into our house through gaps in the wall, under the floors?
I don't know where to put down traps/poison. The only thing I can think is to take up some floorboards in our bedroom (where I hear most of the noises) and do it that way, but not desperately keen to have open voids in the floor. I have a humane trap baited to tempt them out, located where the choccy coin was gnawed, but it's never been tripped. DS2 has autism and I'm not at all keen on poison lying around, plus there's the dopey cat to worry about too. Am fine about using snap traps (sorry) if it means nipping this in the bud.
Sorry, epic!