Curiously, rats and mice like to travel the same path every day, which can leave noticeable evidence of their trails. If you notice what looks like a rodent trail, be very cautious. Rats love to touch things (biologists call this thigmophilic, meaning "touch-loving"), and they use walls to help navigate.
As they run along walls, they tend to leave dark grease marks. You may also notice fecal droppings along the paths of rodents and pests. If the rental unit is dusty, you might be able to see footprints and trail-paths in the dust. Rats often drag their tails through their urine, which also leaves a trail.