Having seen both Mouse and Rat droppings, what you've described sounds like Mice. Rat Droppings are considerably bigger about 1-2 cms long.
If you live in a city like London, it will be difficult to find a property, particulalrly the older ones, that don't have Mice. They live between the walls and under the floorboards.
I kept my flat spotless, but still found droppings behind the fridge and in ground level cupboards. I even found some in my bed once. I agree it was a terrifying thing and I just got the Pest Control people to come in and lay poison everywhere.
They tend to then just go back to their nests and die, however, I did find one lying dead between the bath and toilet. It was horrible having to dispose of it's body,which was still soft.
Mice can get through gaps the width of a pencil, so you would need to seal every gap in the whole property. Rat's tend not to come in the house, unless you have a broken breeze block in the wall, which will allow them entry.